geometric methods in classical field theory and - icmat

Departamento de MatemáticasFacultad de CienciasUniversidad Autónoma de Madrid

Geometric Methods in

Classical Field Theory and

Continuous Media

Cédric M. Campos

Tesis doctoral dirigida por Manuel de León Rodríguez y David Martín de Diego

Madrid, Julio 2010

Departamento de MatemáticasFacultad de CienciasUniversidad Autónoma de Madrid

Métodos Geométricos en

Teoría Clásica de Campos y

Medios Continuos

Cédric M. Campos

Tesis doctoral dirigida por Manuel de León Rodríguez y David Martín de Diego

Madrid, Julio 2010

Contents

Prólogo vii

Agradecimientos ix

Introducción xv

Introduction xix

1 Mathematical background 1

1.1 Distributions and connections . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Multivectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 The geometry of the tangent bundle . . . . . . . . . . . . . . . . . . . . . 41.4 Symplectic geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4.1 The Gotay-Nester-Hinds algorithm . . . . . . . . . . . . . . . . . 81.5 Cosymplectic geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.6 Multisymplectic geometry . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Classical Mechanics 15

2.1 The Lagrangian formalism for autonomous systems . . . . . . . . . . . . 152.1.1 Variational approach . . . . . . . . . . . . . . . . . . . . . . . . . 162.1.2 Geometric formulation . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2 The Hamiltoniam formalism for autonomous systems . . . . . . . . . . . 192.3 The Legendre transformation . . . . . . . . . . . . . . . . . . . . . . . . 202.4 The Tulczyjew's triple . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3 Classical Field Theory 23

3.1 Jet bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.1.1 Prolongations, lifts and contact . . . . . . . . . . . . . . . . . . . 253.1.2 The vertical endomorphisms . . . . . . . . . . . . . . . . . . . . . 283.1.3 Partial Dierential Equations . . . . . . . . . . . . . . . . . . . . 293.1.4 The Dual Jet Bundle . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.2 Classical Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.2.1 The Lagrangian Formalism . . . . . . . . . . . . . . . . . . . . . . 333.2.2 The Hamiltonian Formalism . . . . . . . . . . . . . . . . . . . . . 423.2.3 The Legendre transformation . . . . . . . . . . . . . . . . . . . . 483.2.4 The Skinner and Rusk formalism . . . . . . . . . . . . . . . . . . 50

v

vi CONTENTS

4 Higher Order Classical Field Theory 61

4.1 Higher Order Jet bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.1.1 Prolongations, lifts and contact . . . . . . . . . . . . . . . . . . . 634.1.2 On the denition of vertical endomorphisms . . . . . . . . . . . . 704.1.3 Partial dierential equations . . . . . . . . . . . . . . . . . . . . . 714.1.4 Iterated jet bundles . . . . . . . . . . . . . . . . . . . . . . . . . . 724.1.5 The kth Dual Jet Bundle . . . . . . . . . . . . . . . . . . . . . . . 74

4.2 Higher Order Classical Field Theory . . . . . . . . . . . . . . . . . . . . 784.2.1 Variational Calculus . . . . . . . . . . . . . . . . . . . . . . . . . 784.2.2 Variational Calculus with Constraints . . . . . . . . . . . . . . . . 814.2.3 The Skinner-Rusk formalism . . . . . . . . . . . . . . . . . . . . . 834.2.4 Constraints within the Skinner-Rusk Formalism . . . . . . . . . . 984.2.5 Hamilton-Pontryagin Principle . . . . . . . . . . . . . . . . . . . . 1064.2.6 The space of symmetric multimomenta . . . . . . . . . . . . . . . 107

5 Conclusions and future work 119

5.1 Geometric integrators for higher-order eld theories . . . . . . . . . . . . 1205.2 Space+Time Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . 1225.3 Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1235.4 Continuous Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

5.4.1 The Constitutive Equation . . . . . . . . . . . . . . . . . . . . . . 1255.4.2 Uniformity and Homogeneity . . . . . . . . . . . . . . . . . . . . 1275.4.3 Unisymmetry and Homosymmetry . . . . . . . . . . . . . . . . . 131

A Multi-index properties 137

B Groupoids and G-structures 141

B.1 Groupoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141B.2 G-structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

Prólogo

Este trabajo recopila la investigación desarrollada y los resultados obtenidos durantemis cuatro años como becario predoctoral en el Instituto de Ciencias Matemáticas (yanteriormente a su formación, en el Instituto de Matemáticas y Física Funtamental). Sudefensa tendrá lugar en la Universidad Autónoma de Madrid con objetivo de obtener elgrado de Doctor en Matemáticas. La dirección ha sido llevada a cabo por Manuel de LeónRodríguez, Profesor de Investigación y Director del Instituto de Ciencias Matemáticas,y David Martín de Diego, Investigador Cientíco de la misma institución. En cuanto atutor he contado con Rafael Orive Illera, Profesor Titular de la Universidad Autónoma deMadrid. De la tarea de lector se ha encargado Marco Castrillón López, Profesor Titularde la Universidad Complutense de Madrid.

La monografía versa sobre teoría clásica de campos de orden superior. El lector podráencontrar en sus capítulos iniciales una revisión de algunos de los hechos conocidos enmecánica clásica y teoría clásica de campos (de primer orden). En los capítulos nales,se expone la parte original de la memoria con la extensión de estas teorías a camposclásicos de orden superior, centrándose en la problemática de un formalismo canónicohamiltoniano. Algunos ejemplos son propuestos con el n de facilitar la comprensión yanálisis de los resultados obtenidos.

Se ha pretendido dar una organización gradual y un tratamiento unicado de la ma-teria de tal manera que pueda ser usada en posibles desarrollos futuros.

Cédric M. CamposMadrid, 27 de julio de 2010

vii

viii PRÓLOGO

Agradecimientos

Hace ya varios años, cuando decidí empezar los estudios de doctorado, ya no con el n deobtener el grado de doctor, sino más bien con el simple objetivo de continuar lo que habíainiciado durante mis estudios de licenciatura, no habría podido imaginar los derroterosque tomaría mi vida ni mi carrera académica. No deja de ser un hecho natural e inherenteal devenir del tiempo, pero aún así me sorprende. Después de recorrer este camino, me doycuenta de que una tesis doctoral supone mucho más que la acumulación de conocimientosy el desarrollo intelectual que se le asocia, lo que de por sí no es poco. Al llegar al nalde esta etapa de mi vida, veo como he crecido a nivel personal y profesional, emocionale intelectual. Todo ello no ha sido gratuito, sino gracias a la contribución de las muchaspersonas con las que me he encontrado a lo largo de este tiempo. Como digo, han sidonumerosas y sus aportaciones cuantiosas a la par que diversas, desde señalarme un detalleclave en un problema teórico, a ofrecerme apoyo en una situación comprometida. A todasellas les estoy enormemente agradecido y, sin lugar a dudas, las tendré presentes siempreque recorra las lineas de estas páginas. Por ello, les debo al menos dejar aquí escrito susnombres. Quisiera poder dar gracias personalmente a cada una de ellas, recordar cadacara y cada instante, pero temo fallar. De todas formas, voy a enfrentarme a tan arduatarea.

En primer lugar, quiero agradecer a quiénes han estado conmigo en el día a día,alimentando mi mente con conversaciones cientícas, políticas, losócas o religiosas,siempre acompañadas de buen humor. Ha sido un placer y un honor compartir mis horasde trabajo con todos ellos, pues he disfrutado enormemente al estar rodeado de un grupoque considero parte de la élite cientíca española, a la cual desde luego no pertenezco. Sinduda alguna, echaré de menos el reunirnos a comer en la cafetería del CSIC. De entre ellos,quisiera destacar a Manuel y David, mis directores de tesis. Sin ellos, todo esto no seríaposible. Me dieron la oportunidad (y no sólo una) de embarcarme en este proyecto, medieron problemas a los que enfrentarme (en más de un sentido) y medios para resolverlos.Se han convertido en más que mentores para mí. También un espacial agradecimiento ami compañero de despacho y gran amigo, Ángel. Su rapidez de respuesta nunca dejará desorprenderme, pese a que desgasta la partícula negativa. Ojalá su fútbol tuviese la mismavelocidad. Además, estuvo conmigo en los buenos momentos, en los malos y los peores.Por todo esto, siempre tendrá mi reconocimiento. Tampoco quiero olvidarme de los demásmiembros de la plantilla, todos ellos con algo realmente especial. Gracias a Ana (y sustapers), Ana María, Carolina (enemiga de las impresoras), Dan (por su introducción ala cultural norte-americana y su fe en la selección), David (aka RyC One, mi discípuloubuntero y ex-compañero de habitación), Diego (por los chicles), Emilio (con bici, perosin perro ni auta), Fernando (actual compañero de habitación hasta que la Red o unaRamón y Cajal nos separe), José Antonio (único linuxero convencido, pero estancado enDebian), Luis, María (mi amor en el despacho), Mario (gran escalador), Marta (no tan

ix

x AGRADECIMIENTOS

siniestra), Paco, Paz (con su eterna aura primaveral que nos aportaba juventud a todos)y Roberto (artista redomado, pozo de conocimiento, compañero de viajes a Valencia).No me olvido de los que cruzaron el charco y estuvieron aquí, gracias a Poi (por tantascosas, pero especialmente por los paseos a Malasaña o al centro después de trabajar y lassalidas al Yesterday los nes de semana), a Sebastián y Cecilia (por las argentinadas)y Viviana (por ese humor negro que tanto la hace brillar). Aprovecho este párrafo paraagradecer por su trabajo, por un lado, a mis colaboradores Marcelo y Joris y, por otrolado, a Marco, que ha dedicado su tiempo en leer esta tesis, y a Rafa, por acceder a sermi tutor.

Mi trabajo no sólo se ha visto beneciado de mi labor en el CSIC, sino también delos congresos a los que he asistido y de las personas con las que allí me he encontrado.Esto ha sido, sin lugar a dudas, gracias a la Red de Geometría, Mecánica y Control ysus miembros. Mi trabajo se basa en el de muchos de ellos y no podían faltar aquí susnombres. Difícil sería explayarme aquí individualmente y con detalle, pues la virtud deeste grupo nace de su unión e intercambio de ideas. Mi agradecimiento a todos ellos porsu recibimiento, los buenos consejos y su experiencia, me han sido de gran ayuda. Graciasa Edith, Juan Carlos, David, Diana, Elisa, Miguel C., Narciso, Xavi, María, Eduardo,Jesús, Pepín, Andreu, Javi, Modesto, Silvia, Luis y Miguel.

En ocasiones, las largas horas de trabajo mental se hacían pesadas y me era necesarioliberar la energía acumulada (además de mis frustraciones doctorales). Esto lo hacíaen los intensos partidos de fútbol sala de los jueves jugados en el pabellón deportivo dela Autónoma. Así que agradezco a Diego el haberme invitado insistentemente a dichoseventos y felicitarle por su más que buen hacer en el terreno de juego. A todos losdemás, compañeros y contrincantes, los buenos ratos que he pasado allí. Los golpes hansido duros, pero los goles marcados bien valían la pena. Gracias a Ángel (por cansar aladversario chupándo bola), Dani (subido de Cola-Cao), Eugenio (porterazo), Javi (losdos, a uno por los pases, al otro por la rivalidad), Jorge (siempre tirando de su equipo),Pablo (un maestro), Rafa (siempre decidido), Raúl (los dos), Rubén (generoso), Sasha(aka la Rubia, por el cuerpo a cuerpo), Samuel (lleno de energía), Saúl (por esas patadas)y Wagner. Me atrevo a decir que, pese a mi nula aptitud para este deporte, algo heaprendido y, sin duda, ha sido por ellos.

Además de haberme nutrido intelectualmente en el ámbito laboral, también lo he hechoen mi propia casa. Compartir piso tiene, según muchos, numerosos inconvenientes, peroson escasos ante todas sus ventajas, ante la experiencia de convivir y compartir. Peroello apenas es una pequeña parte de lo que he recibido. Mis compañeros (y allegados)han sido una fuente de riqueza interior, diversidad cultural, conanza, desahogo, risasy diversión. Me siento más que afortunado por haber disfrutado de todo esto día trasdía, de las charlas en la cena, las películas en familia, las salidas en grupo, etc. Detodos los compañeros que han pasado por Mariano de Cavia, destaco mi especial afectoy agradecimiento a David (aka zupermercadona), Julie (e Isabella), Mariana y Enrico (ySara). Tampoco me quiero olvidar de los demás: Marie, Soe, Tom, Andrew, Chris, Will,Je, Mariana (Ángela, Carlos, Diana, Mara y Marcela), Sergio, Viviana, Juanpi y So yZack (y Virginia). No me quiero olvidar de Charo, reina de La Casa de la Charidad, y aGregor por las siempre agradables conversaciones antes de subir al piso. Y por supuesto,a la gente del Rincón: Marcelino, Natalia, Dioni y Manolo, por cientos (o miles) de cañasbien servidas y otras tantas discusiones de bar.

Aparte de en mi casa y en el campo de fútbol, otro lugar de esparcimiento ha estado

xi

en las vastas tierras de Tyria, un mundo virtual más allá de este Mundo Real. Mi alterego (Ashaka Enora) y yo, queremos darle las gracias a mis compañeros de aventuras porlas horas de entretenimiento, las largas charlas y, sobretodo, por el apoyo en los momen-tos difíciles. Gracias a los de Alcorcón(Bronx) David (aka Adnon Mizzrim), Guille (akaKratio Trase), Jorge (aka Demona Almasalvaje) y Sergio (aka Doua Ayame), y a los deValencia, Willy (aka Sir Talent), Xing Chi (aka Pie Mojado) e Yvan (aka Quebrantahue-sos). También incluyo aquí por pertenecer al mundo del bit a Ernestín, uno de los pocosque me entiende. Gracias por los tú aquí, yo allá, y las cañitas en Treviño.

Cuatro años son tiempo y, como dicen, things happen (ocurren cosas), algunas no muybuenas. Afortunadamente uno encuentra buenos profesionales para lidiar con estas. Porun lado, mi agradecimiento al equipo de sioterapeutas del Hospital Gregorio Marañón.Gracias a Miriam por recuperar mis manos y a sus compañeros (y demás pacientes) Gema,José Luis, Cristina, María José y Rubén, por hacerme más llevadera la rehabilitación.Por otro lado, gracias a Carmen por enseñarme a andar, por su profesionalidad y ayuda.

Por último y, como dicen en estos casos, no por ello menos importante, gracias a migente de Valencia: a los de siempre, Ana (y Geraldo) y Ali (y Jose), por su constancia,ánimo y conanza; a la gente del gimnasio, Alfredo y Vicente, por su apoyo; a los de lacarrera, Andrea, Elvira, María Dolores, Xing Chi, Caja, Juan y Olga; a mis tíos y primos,por ser tan ruidosos; a mi querido hermano Yvan, por cuidarme cuando yo no podíahacerlo; a mi padre; y a las mujeres de mi vida, mi madre por su apoyo incondicional, miabuela por sus abrazos, y Laura por su conanza, apoyo, amor y eterna paciencia.

Antes de nalizar, quisiera expresar mi reconocimiento a ciertas personas a las cualesdebo buena parte de mi trabajo pero que, sin embargo, ni siquiera conozco. No sé tratade grandes cientícos de antaño, sino gente del aquí y ahora que con su labor han fa-cilitado la mía (y la de muchos más). Con esto sé que sello mi fama de geek/freakpara siempre, pero bien se lo merecen. Para redactar esta tesis, no sólo en cuanto alesfuerzo de redacción per se, sino también en cuanto a su desarrollo, he hecho uso diariode varias herramientas: GNU/Linux, Emacs, LATEX, Google, Wikipedia, etc. Por ello mireconocimiento a Richard Stallman, fundador del movimiento de Software Libre, desar-rollador del editor de macros Emacs y creador (junto con Linus Torwalds) del sistemaoperativo GNU/Linux; a Lawrence Lessig, precursor del movimiento de Cultura Libre yfundador de la iniciativa Creative Commons; por ende, a toda la comunidad de Softwarey Cultura Libre por desarrollar y crear en benecio de todos; a Donald Knuth y LeslieLamport, creadores de TEXy LATEX, sistemas de composición (especializados en textoscientícos); a Jimmy Wales y Larry Sanger, creadores de la Wikipedia, siempre tan útilpara aclarar conceptos; a Larry Page y Sergey Brin, creadores del buscador Google, sin elcual no encuentro ni mis llaves; y a Mark Shuttleworth, fundador de Canonical, empresadesarrolladora de Ubuntu, mi distribución favorita de GNU/Linux, y a toda la comunidadde Debian/Ubuntu por su ayuda, esfuerzo y contribución.

Gracias.

xii AGRADECIMIENTOS

Generamos conocimiento.Dr. Ángel Castro

Introducción

Una teoría de campos es una teoría física que describe como uno o más campos físicosinteractúan con la materia. Un campo físico puede ser entendido como una asignacióncontinua de una magnitud física en cada punto del espacio y el tiempo: por ejemplo,la velocidad de un uido, el electromagnetismo o incluso la gravedad. Estos son ejemp-los de campos macroscópicos o clásicos en contraste a los microscópicos o cuánticos.Nos centraremos en los primeros. En cierto sentido, la teoría clásica de campos es unageneralización de la mecánica clásica, en la cual el único campo es la linea temporal.

Desde un punto de vista matemático, los campos clásicos pueden ser descritos comosecciones φ de un brado π : E → M . El marco se completa introduciendo una funciónque abarca la dinámica del sistema físico, la función lagrangiana. Para una teoría clásicade campos, esto es una función L : J1π → R, donde J1π es el brado de jets de orden unode π. Este brado de jets ofrece una descripción geométrica de las derivadas parcialesde las coordenadas bradas de E con respecto a las de M , donde una sección es jada.Buscamos pues aquellas secciones φ del brado π : E → M , los campos, que extremizanel funcional

AL(φ,R) =

∫R

L(j1φ)η,

donde η es una forma de volumen prejada (se da por supuesto que M es orientable),R ⊆M es una región compacta de M y j1φ es la primera prolongación jet de φ.

Uno de los resultados más básicos del cálculo variacional es la construcción a partirdel funcional anterior de un conjunto de ecuaciones en derivadas parciales, las ecuacionesde Euler-Lagrange

∂L

∂uα− d

dxi∂L

∂uαi= 0,

las cuales deben ser satisfechas por cualquier extremal suave. Más interesante, la propiedadde extremización del problema no depende de la elección particular del sistema de coor-denadas (hecho que notó J. L. Lagrange durante sus estudios de mecánica analítica), portanto debe ser posible escribir las ecuaciones de Euler-Lagrange de forma intrínseca.

La interpretación geométrica de las ecuaciones de Euler-Lagrange se realiza por mediode la así llamada forma de Poincaré-Cartan ΩL. Esta forma está construida usando lageometría del brado de jets y también está relacionada con el trasfondo variacional[97]. Usando esta forma, es posible escribir la ecuaciones de Euler-Lagrange de formaintrínseca. Es más, φ satisface las ecuaciones de Euler-Lagrange (es decir, es un puntocrítico de la acción AL) si y sólo si

(j1φ)∗(iV ΩL) = 0, para todos los vectores tangentes V en TJ1π.

Además, esta forma juega un papel importante en la conexión entre las simetrías y lasleyes de conservación (see [53]).

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xvi INTRODUCCIÓN

Otra manera de describir la evolución de los campos es introduciendo una funcióndinámica el dual a J1π, esto es, una función hamiltoniana H : J1π† → R, donde J1π†

es el dual extendido del brado de jets de primer orden de π. Entonces, la dinámicadel sistema viene descrita gracias a las soluciones de las bien conocidas ecuaciones deHamilton

∂H

∂uα= −∂p

iα

∂xiy

∂H

∂piα=∂uα

∂xi,

las cuales son extremales para el principio variacional dado en J1π† (véanse [31, 61, 115,134, 147]).

La relación entre estos dos marcos, el formalismo lagrangiano y el hamiltoniano, es de-scubierto por la transformada de Legendre. Dado un lagrangiano L : J1π → R, podemosdenir el mapa LegL : J1π → J1π†. Esta función tiene interesantes propiedades comoenviar las soluciones de las ecuaciones de Euler-Lagrange a soluciones de las ecuacionesde Hamilton, o bien retrotraer la (m+1)-forma de Cartan ΩH de J1π† a la (m+1)-formade Poincaré-Cartan J1π (véanse [35, 61, 134, 147]). Es más, cuando L es regular, esto escuando su hessiano (

∂2L

∂vα∂vβ

)es regular, la transformada de Legendre LegL es un difeomorsmo local en su imagen, lacual es a su vez difeomorfa al dual reducido del brado de jets de primer orden, J1π.

En la actualidad, se posee una muy buena comprensión de las teorías de campos deprimer orden. Pero muchos lagrangianos que aparecen las teorías de campos son de ordensuperior (como por ejemplo en elasticidad o gravitación), por tanto es de sumo interésencontrar un marco completamente geométrico también para estas teorías de campos, estocuando uno considera una función lagrangiana L : Jkπ → R, donde Jkπ es el brado dejets de orden k de π. Durante las últimas décadas del pasado siglo, han habido diferentesestudios e intentos para denir de manera global e intrínseca el cálculo variacional deorden superior en varias variables. Los objetivos principales son describir las ecuacionesde Euler-Lagrange asociadas para secciones del brado, derivar las formas de Poincaré-Cartan como versión intrínseca las ecuaciones anteriores, y construir transformadas deLegendre adecuadas que nos permitan escribir estas ecuaciones en el marco hamiltoniano(véanse, por ejemplo, [4, 6, 65, 66, 89, 94, 106, 113, 63, 62, 140] para más información).

El marco geométrico estándar de la teoría de campos de orden superior se inicia conla búsqueda de extremales del funcional

AL(φ,R) =

∫R

L(jkφ)η,

donde como antes η es una forma de volumen prejada, R ⊆M es una región compactay jkφ es la prolongación k-jet de φ. El calculo variacional establece que los extremales deesta acción integral deben satisfacer las ecuaciones de Euler-Lagrange de orden superior

k∑|J |=0

(−1)|J |d|J |

dxJ∂L

∂uαJ=

∂L

∂uα− d

dxi∂L

∂uαi+

d2

dxij∂L

∂uα1i+1j

− · · ·+ (−1)kdk

dxJ∂L

∂uαJ= 0,

las cuales son un conjunto de ecuaciones en derivadas parciales en J2kπ. Al igual queen el caso de primer orden, estas ecuaciones no dependen de la elección de coordenadas.

xvii

Por tanto, uno puede llegar a preguntarse si existe un objeto canónico que describageométricamente este conjunto de ecuaciones y sus soluciones. En tal caso, debería deser una forma de Poincaré-Cartan de orden superior.

La situación está bien establecida para el caso de una variable dependiente (mecánicade orden superior) y para el caso de primer orden [85, 91, 93, 105]. En este último caso,la expresión típica de la forma de poincaré-Cartan asociado en mecánica clásica a unlagrangiano L : J1π → R puede ser escrita como S∗(dL) + Ldt, donde S∗ es el adjuntodel endomorsmo vertical actuando sobre 1-formas. Con el objetivo de generalizar esteconcepto a teorías de campos de orden superior, uno necesita denir una aplicación de las1-formas (la diferencial de L) a m-formas e incorporar de manera global las derivadas deorden superior. Esta es una de las razones para el grado de arbitrariedad en la deniciónde la forma de Cartan para funciones lagrangianas L : Jkπ → R, con k > 1 y dimM > 1.En otras palabras, habrán diferentes formas de Cartan denidas a partir de la mismafunción que denan una formulación intrínseca de las ecuaciones de Euler-Lagrange. Larazón principal de este problema es la conmutatividad de las derivadas parciales iteradas.Por tanto, la forma de Cartan es única si (y sólo si) bien k o bien m es igual a uno.

En la literatura, encontramos diferentes aproximaciones para jar la forma de Cartanen teorías de campos de orden superior. Un trato directo es la aproximación de Aldaya yAzcárraga [4, 6]. Otro punto de vista es el de Arens [8], que consiste en inyectar el bradode jets Jkπ en un brado de jets de orden 1 apropiado, con la introducción de numerosasvariables dentro de una teoría de multiplicadores de Lagrange. Desde un punto de vistamás geométrico, García y Muñoz describen un método global para la construcción deformas de Poincaré-Cartan en el calculo de variaciones de orden superior de espaciosbrados por medio de conexiones (see [88, 89]). En particular, construyen formas deCartan que dependen de la elección de dos conexiones, una conexión lineal en la base y otraconexión lineal en el brado vertical V π. Más tarde, Crampin y Saunders [140] proponenel uso de operadores análogos a la estructura casi tangente canónicamente denida en elbrado tangente de una variedad de conguración dada M para la construcción globalde formas de Poincaré-Cartan; este operador depende de la forma de volumen elegida enla base.

En esta monografía, propones un camino alternativo, evitando el uso de estructurasadicionales y trabajando únicamente con objetos intrínsecos del lado lagrangiano y delhamiltoniano. Los resultados pueden encontrase publicados en [24, 25, 26, 27] (para unpunto de vista complementario, se sugiere el trabajo de Vitagliano [152]). Con vistas atratar sistemas lagrangianos singulares, Skinner y Rusk construyen un sistema hamilto-niano en la suma de Whitney TQ ⊕ T ∗Q de los brados tangente y cotangente de unavariedad de conguración Q. La ventaja de su acercamiento yace en el hecho de quela condición de segundo orden de la dinámica es satisfecha automáticamente. Esto noocurre en el lado lagrangiano de la formulación de Gotay y Nester, donde la condition desegundo orden debe de ser considerada tras la implementación del algoritmo de ligaduras(ver [100, 101, 102]), aunque otros formalismos incluyen esta condición de segundo ordendesde el principio (ver [34, 36]).

En teorías de campos de orden superior, empezamos con un lagrangiano denido enJkπ. Consideramos el brado πW,M : W → M , donde W = Jkπ ×Jk−1π Λm

2 (Jk−1π)es un producto brado, el espacio de velocidades y momentos. En W construímos unaforma premultisimpléctica haciendo el pull back de la forma multisimpléctica canónicade Λm

2 (Jk−1π), y denimos un formalismo hamiltoniano conveniente gracias al pairing

xviii INTRODUCCIÓN

natural canónico y la función lagrangiana dada. Las soluciones de la ecuación de camposson entendidas como secciones integrales de conexiones de Ehresmann en el brado πW,M :W → M . En este espacio, obtenemos una expresión global, intrínseca y única de unaecuación tipo Cartan para las ecuaciones de Euler-Lagrange para teorías de campos deorden superior. Adicionalmente, obtenemos algoritmo de ligaduras. Nuestro esquema esaplicado a diferentes ejemplos para ilustrar el método.

A parte de la carencia de ambigüedad inherente a nuestra construcción, vale la penaenfatizar que este formalismo se puede extender fácilmente a teorías de campos de ordensuperior con restricciones o problemas de control óptimo en ecuaciones en derivadas par-ciales. En este sentido, obtenemos una descripción unicadora y geométrica de ambostipos de sistemas, con posibles aplicaciones futuras a teorías de reducción por simetrías yla construcción de métodos numéricos que preserven la estructura geométrica (ver [116]).Por tanto, introducimos restricciones en el marco de trabajo, las cuales están represen-tadas geométricamente como una subvariedad C de Jkπ. En otras palabras, imponemosrestricciones en el espacio de secciones donde la acción está denida. El formalismo in-troducido en [27] es adaptado al caso de teorías de campos restringidas, derivando así unmarco intrínseco de las ecuaciones de Euler-Lagrange restringidas. Para la descripcióngeométrica, inducimos una subvariedad W C

0 de W usando las restricciones dadas por C.Algunos ejemplos son dados para ilustrar la teoría, la cual está recogida en [26].

El Capítulo 1 recopila la notación utilizada a lo largo de la monografía así comoel fondo matemático necesario: distribuciones, las diferentes geometrías simplécticas, laestructura del brado tangente, etc. También contiene un esquema del algoritmo deGotay-Nester-Hinds.

El Capítulo 2 es una somera revisión de la mecánica clásica. Describe los principalesresultados de la teoría desde el lado lagrangiano y el hamiltoniano.

El Capítulo 3 es una breve introducción a la teoría clásica de campos. Desarrolla lateoría (generalmente sin pruebas) en los diferentes formalismos, el lagrangiano y el hamil-toniano, y los diferentes posibles acercamientos, el variacional y el geométrico. Tambiénmuestra la relación entre ellas e introduce el formalismo de Skinner y Rusk para teoríasde campos.

El Capítulo 4 está dedicado al estudio de la teoría clásica campos clásicos de ordensuperior. El lector podrá encontrar una primera generalización de los principales objetosgeométricos de la teoría de primer orden, señalando las causas de la ambigüedad inherentea la teoría de orden superior. En adelante, el capítulo se centra en la resolución de estaambigüedad por medio del formalismo de Skinner y Rusk. También introduce restric-ciones en el esquema. Finalmente, hay una presentación de algunos resultados parcialesen la reducción de la arbitrariedad en el espacio de soluciones de la teoría.

Por último, el Capítulo 5 expone un resumen de los principales resultados obtenidosa lo largo de mis estudios, junto con algunas conclusiones y los trabajos futuro que seinician con este tratado.

Introduction

A eld theory is a physical theory that describes how one or more physical elds interactwith matter. A physical eld can be thought of a continuous assignment of a physicalquantity at each point of space and time: For instance, the velocity of a uid, electro-magnetism or even gravitation. These are macroscopic or classical eld examples incontrast to microscopic or quantum ones. We will focus on the former. In some sense,classical eld theory is a generalization of classical mechanics, in which the only eld isthe time line.

From the mathematical point of view, classical elds may be described by sections φof a ber bundle π : E → M . The picture is completed by introducing a function thatencompasses the dynamics of the physical system, the Lagrangian. For rst order eldtheories, it is a function L : J1π → R, where J1π is the rst-jet bundle of π. This jetbundle gives a geometrical description of the partial derivatives of the ber coordinatesof E with respect to those of M , where a section is xed. We then look for those sectionsφ of the ber bundle π : E →M , the elds, that extremize the functional

AL(φ,R) =

∫R

L(j1φ)η,

where η is a xed volume form (it is assumed that M is orientable and oriented), R ⊆Mis a compact region of M and j1φ is the 1st-jet prolongation of φ.

The most basic result on variational calculus is the construction from the above func-tional of a set of partial dierential equations, the Euler-Lagrange equations

∂L

∂uα− d

dxi∂L

∂uαi= 0,

which must be satised by any smooth extremal. More interesting, the property ofextremizing the problem does not depend on the particular chosen coordinate system(fact noted by J. L. Lagrange during his studies of analytical mechanics), therefore itmust be able to write the Euler-Lagrange equations in an intrinsic way.

The geometric interpretation of the Euler-Lagrange equations is done by means of theso-called Poincaré-Cartan form ΩL, which is an (m + 1)-form (dimM = m) univocallyassociated to the Lagrangian. This form is constructed using the geometry of the jetbundle and it is also related with the variational background [97]. Using this form, itis possible to write down the Euler-Lagrange equations in an intrinsic way. Indeed, φsatises the Euler-Lagrange equations (that is, it is a critical point of the action AL) ifand only if

(j1φ)∗(iV ΩL) = 0, for all tangent vector V in TJ1π.

Moreover, this form plays an important role in the connection between symmetries andconservation laws (see [53]).

xix

xx INTRODUCTION

Besides, another way to describe the evolution of the elds is by introducing a dynami-cal function in the dual side of J1π, that is, by introducing the HamiltonianH : J1π† → R,where J1π† is the extended dual rst-jet bundle of π. Then, the dynamics of the systemis described by means of the solutions of the well known Hamilton's equations

∂H

∂uα= −∂p

iα

∂xiand

∂H

∂piα=∂uα

∂xi,

which are extremals of a variational principle given in J1π† (see [31, 61, 115, 134, 147]).The relation between these two pictures, the Lagrangian and the Hamiltonian for-

malisms, is unveiled by the Legendre transformation. Given a Lagrangian L : J1π → R,we may dene a mapping LegL : J1π → J1π†. This function has interesting propertieslike it maps the solutions of the Euler-Lagrange equation to solutions of the Hamilton'sequations or it pulls back the Cartan (m+ 1)-form ΩH of J1π into the Poincaré-Cartan(m + 1)-form of J1π (see [35, 61, 134, 147]). Moreover, when L is regular, that is whenits Hessian (

∂2L

∂vα∂vβ

)is regular, the Legendre map LegL is a local dieomorphism into its image, which is inits turn dieomorphic to the reduced dual rst-jet bundle J1π.

So far, the rst-order case of eld theories is pretty well understood. But many of theLagrangians which appear in eld theories are of higher order (as for instance in elasticityor gravitation), therefore it is interesting to nd a fully geometric setting also for theseeld theories, that is when one considers a Lagrangian function L : Jkπ → R, where Jkπis the kth-order jet bundle of π. During the last decades of the past century, there havebeen dierent studies and attempts to dene in a global and intrinsic way the higher-order calculus of variations in several independent variables. The main objectives are todescribe the associated Euler-Lagrange equations for sections of the ber bundle, to derivePoincaré-Cartan forms for use in intrinsic versions of the above equations, and to constructadequate Legendre maps which permit to write the equations in the Hamiltonian side (see,for instance, [4, 6, 65, 66, 89, 94, 106, 113, 63, 62, 140] for further information).

The standard geometric framework of higher-order eld theories starts by looking forthe extremals of the functional

AL(φ,R) =

∫R

L(jkφ)η,

where as before η is a xed volume form, R ⊆M is a compact region and jkφ is the k-jetprolongation of φ. Variational calculus states that the extremizers of this integral actionmust satisfy the higher-order Euler-Lagrange equations

k∑|J |=0

(−1)|J |d|J |

dxJ∂L

∂uαJ=

∂L

∂uα− d

dxi∂L

∂uαi+

d2

dxij∂L

∂uαij− · · ·+ (−1)k

dk

dxJ∂L

∂uαJ= 0,

which is a set of partial dierential equations in J2kπ. As in the rst order case, theseequations do not depend on the chosen coordinates. Thus, one may wonder if there is acanonical object that describes geometrically this set of equations and their solutions. Insuch a case, it should be a higher-order Poincaré-Cartan form.

xxi

The situation is well established for the case of one independent variable (higher ordermechanics) and for the case of rst order calculus of variations [85, 91, 93, 105]. In thislast situation, the typical expression of the Poincaré-Cartan form associated in classicalmechanics to a Lagrangian L : J1π → R may be written as S∗(dL) + Ldt, where S∗is the adjoint of the vertical endomorphism acting on 1-forms. In order to generalizethis concept to higher order eld theories, one needs to dene a mapping from 1-forms(the dierential of L) to m-forms and to incorporate in a global way the higher orderderivatives. This is one of the reasons for the degree of arbitrariness in the denition ofCartan forms for Lagrangian functions L : Jkπ → R, with k > 1 and dimM > 1. Inother words, there will be dierent Cartan forms which carry out the same function inorder to dene an intrinsic formulation of Euler-Lagrange equations. The main reasonof this problem is the commutativity of repeated partial dierentiation. Therefore, theCartan form is unique if (and only if) either k or m equals one.

In the literature, we nd dierent approaches to x the Cartan form for higher ordereld theories. A direct attempt is the approach by Aldaya and Azcárraga [4, 6]. Anotherpoint of view is that by Arens [8], which consists of injecting the jet bundle Jkπ to anappropriate rst-order jet bundle by the introduction of a great number of variables intothe theory and Lagrange multipliers. From a more geometrical point of view, García andMuñoz described a method of constructing global Poincaré-Cartan forms in the higherorder calculus of variations in bered spaces by means of linear connections (see [88, 89]).In particular they show that the Cartan forms depend on the choice of two connections, alinear connection on the baseM and a linear connection on the vertical bundle V π. Later,Crampin and Saunders [140] proposed the use of an operator analogous to the almosttangent structure canonically dened on the tangent bundle of a given congurationmanifold M for the construction of global Poincaré-Cartan forms; this operator dependson the chosen volume form on the base.

In this monograph, we propose an alternative way, avoiding the use of additionalstructures, working only with intrinsic objects from both the Lagrangian and Hamiltoniansides. The results main be found published here [24, 25, 26, 27] (for a complementarypoint of view, see the work by Vitagliano [152]). This formalism is strongly based onthe one developed by Skinner and Rusk [142, 143, 144]. In order to deal with singularLagrangian systems, Skinner and Rusk construct a Hamiltonian system on the Whitneysum TQ ⊕ T ∗Q of the tangent and cotangent bundles of the conguration manifold Q.The advantage of their approach lies on the fact that the second order condition ofthe dynamics is automatically satised. This does not happen in the Lagrangian sideof the Gotay and Nester formulation, where the second-order condition problem hasto be considered after the implementation of the constraint algorithm (see [100, 101,102]), besides other formalisms which include the second-order condition from the verybeginning (see [34, 36]).

For higher-order eld theories, we start with a Lagrangian function dened on Jkπ.We consider the bration πW,M : W →M , where W = Jkπ×Jk−1π Λm

2 (Jk−1π) is a beredproduct, the velocity-momentum space. OnW we construct a premultisymplectic form bypulling back the canonical multisymplectic form of Λm

2 (Jk−1π), and we dene a convenientHamiltonian from a natural canonical pairing and the given Lagrangian function. Thesolutions of the eld equations are viewed as integral sections of Ehresmann connectionsin the bration πW,M : W → M . In this space we obtain a global, intrinsic and uniqueexpression for a Cartan type equation for the Euler-Lagrange equations for higher-order

xxii INTRODUCTION

eld theories. Additionally, we obtain a resultant constraint algorithm. Our scheme isapplied to several examples to illustrate our method.

Apart from the lack of ambiguity inherent in our construction, it is worth to empha-size that this formalism is easily extended to the case of higher-order eld theories withconstraints and optimal control problems for partial dierential equations. In this way,we obtain a unied, geometric description of both types of systems, with possible futureapplications in the theory of symmetry reduction and the construction of numerical meth-ods preserving geometric structure (see [116]). Therefore, we introduce constraints in thepicture, which are geometrically dened as a submanifold C of Jkπ. In other words, weimpose the constraints on the space of sections where the action is dened. The formalismintroduced in [27] is adapted to the case of constrained eld theories, deriving an intrinsicframework of the constrained Euler-Lagrange equations. For the geometrical description,we induce a submanifold W C

0 of W using the constraints given by C. Some examples aregiven to illustrate the theory, which appears in [26]

Chapter 1 gathers the notation used along the monograph and the basic mathematicalbackground needed: distributions, the dierent symplectic geometries, the structure ofthe tangent bundle, etc. There is also a sketch of the Gotay-Nester-Hinds algorithm.

Chapter 2 is a short review of Classical Mechanics. It depicts the main results of thetheory from the Lagrangian and the Hamiltonian side.

Chapter 3 is a brief introduction to Classical Field Theory. It develops the theory(generally without proofs) within the dierent formalisms, Lagrangian and Hamiltonian,and the dierent approaches, variational and geometrical. It also shows the relationbetween them and introduces the Skinner-Rusk formalism for eld theories.

Chapter 4 is devoted to the study of Higher-Order Classical Field Theory. The readerwill may nd rst a generalization of the main geometric objects of the rst order theory,pointing out the causes of the ambiguity inherent to the higher-order theory. Then thechapter focuses on resolution of this ambiguity by means of the Skinner-Rusk formalism.It also introduces constraints in the pictures. Finally there is a presentation of the partialresults on the reduction of the arbitrariness in the space of solutions of the theory.

Finally, Chapter 5 exposes a summary of the main results obtained along my studies,together with some conclusions and the future work that starts with this treatise.

Chapter 1

Mathematical background

1.1 Distributions and connections

See [1, 122] for an introduction to the theory of connections.

Denition 1.1. A distribution D of dimension m on a manifold P is an assignment toeach p ∈ P of a vector subspace D(p) ⊆ TpP of dimension m.

1. A distribution D of dimension m is smooth if, for each p0 ∈ P , there exist anopen neighborhood Up0 of p0 and local vector elds Y1, . . . , Ym ∈ X(Up0), such thatY1(p), . . . , Ym(p) span D(p) for every p ∈ Up0 .

2. A submanifold S → P is said to be an integral manifold of a smooth distributionD in TP if TS = D along the points of S. In such a case, D is said to be integrable.

3. A smooth distribution D is involutive if it is stable under the Lie bracket, that is,if [D,D] ⊆ D.

Theorem 1.2 (Frobenius' Theorem). A smooth distribution D is integrable if and onlyif it is involutive.

Denition 1.3. A connection Γ in a ber bundle πP,M : P → M is given by a πP,M -horizontal distribution H in TP , i.e. a distribution H in TP which is complementary tothe vertical one V πP,M , that is

TP = D ⊕ V πP,M ,

where V πP,M(p) = V ∈ TpP : TpπP,M(V ) = 0. This decomposition allow us to dene:

1. The horizontal projector associated to the connection Γ is the linear map h : TP →D dened in the obvious manner.

2. The horizontal lift of a tangent vector X ∈ TM is the unique vector Xh ∈ D thatprojects to X, TπP,M(Xh) = X.

3. If (xi, ya) are bered coordinates on P , then D is locally spanned by the local vectorelds (

∂

∂xi

)h=

∂

∂xi+ Γai (x, y)

∂

∂ya.

The coecients Γai are the Christoel symbols of the connection.

1

2 CHAPTER 1. MATHEMATICAL BACKGROUND

Assume that πQ,M : Q → M and πP,M : P → M are two brations with the samebase manifold M , and that Υ : Q → P is a surjective submersion (in other words, abration as well) preserving the brations, say, πP,M Υ = πQ,M (Diagram 1.1). Let Γ′

be a connection in πQ,M : Q→M with horizontal projector h′.

Q Υ //

πQ,M @@@@@@@@ P

πP,M

M

Figure 1.1: Preserved bration

Denition 1.4. Γ′ is said to be projectable if the subspaces (TqΥ)(D′(q)) are constantalong the bers of Υ, i.e. (Tq1Υ)(D′(q1)) = (Tq2Υ)(D′(q2)) for every q1, q2 ∈ Υ−1(p),p ∈ P .

If Γ′ is projectable, then we dene a connection Γ in the bration πP,M : P → M asfollows: The horizontal subspace at p ∈ P is given by

Dp = (TqΥ)(D′(q)),

for an arbitrary q in the bre of Υ over p. It is routine to prove that D denes a horizontaldistribution in the bration πP,M : P →M .

We can choose bered coordinates (xi, ya, zα) on Q such that (xi, ya) are beredcoordinates on P . The Christoel components of Γ′ are obtained by computing thehorizontal lift (

∂

∂xi

)h=

∂

∂xi+ Γai (x, y, z)

∂

∂ya+ Γαi (x, y, z)

∂

∂zα.

A simple computation shows that Γ′ is projectable if and only if the Christoel compo-nents Γai are constant along the bres of Υ, say Γai = Γai (x, y). In this case, the horizontallift of ∂/∂xi with respect to Γ is just(

∂

∂xi

)h=

∂

∂xi+ Γai (x, y)

∂

∂ya.

Conversely, given a connection Γ in the bration πP,M : P → M and a surjective sub-mersion Υ : Q → P preserving the brations, one can construct connections Γ′ in thebration πQ,M : Q → M which project onto Γ (rst, locally and then globally by meansof a partition of the unity).

The notion of connection in a bration admits a useful generalization to submanifoldsof the total space. Let πP,M : P →M be a bration and N a submanifold of P .

Denition 1.5. A connection in πP,M : P → M along the submanifold N of P consistsof a family of linear mappings

hp : TpP −→ TpN

for all p ∈ N , satisfying the following properties

h2p = hp, kerhp = Vp πP,M ,

for all p ∈ N . The connection is said to be smooth (at) if the distribution imh ⊆ TNis smooth (integrable).

1.2. MULTIVECTORS 3

We have the following.

Proposition 1.6. Let h be a connection in πP,M : P →M along a submanifold N of P .Then:

1. πP,M(N) is an open subset of M .

2. (πP,M)|N : N → πP,M(N) is a bration.

3. There exists an induced true connection ΓN in the bration (πPM)|N : N → πPM(N)with the same horizontal subspaces.

4. ΓN is at if and only if h is at.

Proof. See [55, 50].

1.2 Multivectors

Denition 1.7. Let P be a n-dimensional dierentiable manifold. Sections of Λm(TP )(with 1 ≤ m ≤ n) are called m-multivector elds in P . The set of m-multivector eldsin P is denoted by Xm(P ).

Given X ∈ Xm(P ), for every p ∈ P , there exists an open neighborhood Up ⊂ P andX1, . . . , Xr ∈ X(Up) such that

X =Up

∑1≤i1<...<im≤r

f i1...imXi1 ∧ . . . ∧Xim

with f i1...im ∈ C∞(Up) and m ≤ r ≤ n. Of particular interest are those multivector eldswhose decomposition may be reduced to a single term.

Denition 1.8. A multivector eld X ∈ Xm(P ) is locally decomposable if, for everyp ∈ P , there exists an open neighborhood Up ⊂ P and X1, . . . , Xm ∈ X(Up) such that

X =UpX1 ∧ . . . ∧Xm.

The set of locally decomposable m-multivector elds in P is denoted by Xmd (P ).

Let D ⊆ TP be an m-dimensional distribution. The sections of ΛmD are locallydecomposable m-multivector elds in P .

Denition 1.9. A locally decomposable m-multivector eld X ∈ Xmd (P ) and an m-

dimensional distribution D ⊆ TP are associated whenever X is a section of ΛmD.

If X,X ′ ∈ Xmd (P ) are non-vanishing multivector elds associated with the same dis-

tribution D, then there exists a non-vanishing function f ∈ C∞(P ) such that X ′ = fX.This fact denes an equivalence relation in the set of non-vanishing m-multivector eldsin P , whose equivalence classes will be denoted by D(X).

Theorem 1.10. There is a bijective correspondence between the set of m-dimensionalorientable distributions D in TP and the set of the equivalence classes D(X) of non-vanishing, locally decomposable m-multivector elds X in P .


By abuse of notation, D(X) will also denote them-dimensional orientable distributionD in TP with whom X is associated.

Denition 1.11. An m-dimensional submanifold S → P is said to be an integral mani-fold of X ∈ Xm(P ) if X spans ΛmTS. In such a case, X is said to be integrable.

Note that integrable multivector elds are necessarily locally decomposable.

Denition 1.12. A non-vanishing, locally decomposable m-multivector X ∈ Xmd (P ) is

involutive if its associated distribution D(X) is involutive.

If a non-vanishing multivector eld X ∈ Xmd (P ) is involutive, so is every other in

its equivalence class D(X). Furthermore, by Frobenius' theorem we have the followingresult.

Corollary 1.13. A non-vanishing and locally decomposable multivector eld is integrableif, and only if, it is involutive.

Denition 1.14. Let πP,M : P →M be a ber bundle with dimM = m. A multivectoreld X ∈ Xm(P ) is said to be π-transverse if Λmπ∗(X) does not vanish at any point ofM , hence M must be orientable.

Proposition 1.15. If X ∈ Xm(P ) is integrable, then X is π-transverse if, and only if,its integral manifolds are sections of π : P →M . In this case, if S is an integral manifoldof X, then there exists a section φ ∈ Γπ shuch that S = Im(φ).

For more details on multivector elds and their relation with eld theories, we referto [72, 73].

1.3 The geometry of the tangent bundle

Through this section, Q denotes an n-dimensional smooth manifold. Local coordinatesin Q are denoted (qi), and the induced adapted coordinates of TQ and TTQ are denoted(qi, vi) and (qi, vi, qi, vi), respectively. According to this, vectors v ∈ TqQ and V ∈ Tv(TQ)are respectively of the form

v = vi∂

∂qi

∣∣∣qand V = qi

∂

∂qi

∣∣∣v

+ vi∂

∂vi

∣∣∣v.

If τQ : v ∈ TqQ 7→ q ∈ Q denotes the natural projection of TQ onto Q then, given atangent vector V ∈ Tv(TQ), we have that τTQ(V ) = v. Besides, we also have the followingcoordinate expressions (see Diagram 1.2)

τQ(qi, vi) = (qi), τTQ(qi, vi, qi, vi) = (qi, vi) and TτQ(qi, vi, qi, vi) = (qi, qi).

Denition 1.16. Let v ∈ TqQ be a vector tangent to Q at some point q ∈ Q. Thevertical lift of v at a point w ∈ TqQ is the tangent vector vv

w ∈ Tw(TQ) given by

vvw(f) =

d

dtf(w + tv)

∣∣t=0, ∀f ∈ C∞(TqQ). (1.1)

1.3. THE GEOMETRY OF THE TANGENT BUNDLE 5

TTQ

τTQ

TτQ // TQ

τQ

TQ

τQ // Q

Figure 1.2: The natural projections

Given a smooth function f ∈ C∞(Q),

(TwτQ)(vvw)(f) = vv

w(f τQ)

=d

dt(f τQ)(w + tv)

∣∣t=0

=d

dtf(q)

∣∣t=0

= 0.

Thus, the vertical lift takes values into the vertical ber bundle V τQ ⊂ TTQ. Indeed, foreach w ∈ TqQ, the vertical lift at w,

(·)vw : TqQ −→ Vw τQ ⊂ TwTQ,

is a linear isomorphism. It may also be seen as a morphism X ∈ X(Q) 7→ Xv ∈ Xv(TQ),where Xv(TQ) is the module of vector elds over TQ that are vertical with respect to theprojection τQ. In local coordinates, if v = vi ∂

∂qi|q = (qi, vi) and w = wi ∂

∂qi|q = (qi, wi),

then

vvw = vi

∂

∂vi

∣∣∣w

= (qi, wi, 0, vi)

for the induced adapted local coordinates of TTQ.

Denition 1.17. The vertical endomorphism is the linear map S : TTQ −→ TTQ that,for any vector V ∈ TTQ, gives the value

S(V ) = ((TvτQ)(V ))v, (1.2)

where v = τTQ(V ) ∈ TQ.

In adapted coordinates (qi, vi) of TQ, the vertical endomorphism has the local expres-sion

S = dqi ⊗ ∂

∂vior S(qi, vi, qi, vi) = (qi, vi, 0, qi). (1.3)

Denition 1.18. The Liouville or dilation vector eld is the vector eld ∆ over TQdened by

∆v = (vv)v, (1.4)

for any v ∈ TQ.

In adapted coordinates (qi, vi) of TQ, ∆ is given by

∆ = vi∂

∂vi= (qi, vi, 0, vi). (1.5)

Another way to dene the Liouville vector eld is as the innitesimal generator of the1-parameter group of transformations φt : v ∈ TQ 7→ etv ∈ TQ. This denition caneasily be translated to any vector bundle.


Denition 1.19. A second order vector eld or dierential equation (usually abbreviatedSODE ) is a vector eld X ∈ X(TQ) such that TτQ X = IdTQ.

In adapted coordinates (qi, vi) of TQ, a SODE is a vector eld

X = X i ∂

∂qi+ Y i ∂

∂visuch that X i = vi.

Thus, neither the Liouville vector eld nor the vertical lift of a vector eld are secondorder vector elds. Even though, SODEs are characterized by the equation

S(X) = ∆.

Denition 1.20. Given a smooth curve c : I −→ Q, its (rst) lift to TQ is the smoothcurve c(1) : I −→ TQ such that

(c(1)(t0))(f) =d

dt(f c)

∣∣∣t=t0

.

In local adapted coordinates, c(1) = (ci, dci/ dt).

Proposition 1.21. A vector eld X ∈ X(TQ) is a SODE if and only if the integralcurves of X are lifts of their own projections to Q; that is, if c is an integral curve of X,then

c = (τQ c)(1). (1.6)

The curve c = τQ c : I −→ Q is called a base integral curve of X or a solution of theSODE given by X.

If c : I −→ TQ is an integral curve of a SODE X ∈ X(TQ) locally given by X =(qi, vi, vi, ai) and c : I −→ Q denotes its base integral curve, then

qi = ci, vi =dci

dtand ai =

d2ci

dt2.

Alternatively, the base integral curve c of c satises the system of second order dierentialequations

d2ci/ dt2 = ai(ci, dci/ dt) (intrinsically c(1)(t) = X(c(1)(t))).

1.4 Symplectic geometry

In some sense, symplectic geometry is complementary to Riemannian geometry. WhileRiemannian geometry is based on the study of smooth manifolds that carry a non-degenerate symmetric tensor, symplectic geometry covers the study of smooth manifoldsthat are equipped with a non-degenerate skewsymmetric tensor. Although both haveseveral similarities, by their nature they also show to have strong dierences.

Along this section, V and M respectively denote a real vector space and a smoothmanifold. They do not necessarily have nite dimension.

1.4. SYMPLECTIC GEOMETRY 7

Denition 1.22. Let ω : V × V −→ R be a bilinear map and dene the morphismω[ : V −→ V ∗ by ⟨

ω[(v)|w⟩

= ω(v, w).

We say that ω is weakly (resp. strongly) non-degenerate whenever ω[ is a monomorphism(resp. an isomorphism).

It turns out that, if V is nite-dimensional, weak and strong non-degeneracy coincide.Thus, in this case, we simply use the term non-degenerate.

Proposition 1.23. Let V be a nite-dimensional real vector space and let ω ∈ Λ2V ∗ bea skew-symmetric bilinear map. The following holds,

1. ω is non-degenerate if and only if V is even-dimensional (dimV = 2n) and theexterior nth-power ωn is a volume form on V ;

2. if ω is non-degenerate, then there exists a basis (εi)2ni=1 in V ∗ such that

(ωij) =

(0 I−I 0

),

where ω = ωij εi⊗εj, 0 is the n-by-n null matrix and I is the n-dimensional identity

matrix. Equivalently, ω =∑n

i=1 εi∧ εn+i.

Denition 1.24. A weak (resp. strong) symplectic form on a real vector space V is aweakly (resp. strongly) non-degenerate 2-form ω on V . The pair (V, ω) is called a weak(resp. strong) symplectic vector space.

As before, we avoid the use of the terms weak and strong in the case of nite-dimensional vector spaces.

Example 1.25. Let V be a real vector space of dimension n. Let (ei)ni=1 be a basis of

V and let (εi)ni=1 be its dual counterpart (i.e. εi(ej) = δij). Then, with some abuse ofnotation, ω =

∑ni=1 ε

i∧ ei is a non-degenerate 2-form in V × V ∗. Note that ω does notdepend on the chosen basis (ei)

ni=1 of V . In fact, ω may be dened intrinsically by the

following expression,ω((v1, α1), (v2, α2)) = α2(v1)− α1(v2).

Denition 1.26. Let M be a smooth manifold, a tensor eld ω ∈ Ω2(M) is weakly(resp. strongly) non-degenerate if the bilinear map ωx : TxM × TxM −→ R is weakly(resp. strongly) non-degenerate, for each x ∈M .

Proposition 1.27. Given a tensor eld ω over M of type (0, 2), let ω[ : X(M) −→ Ω(M)be the mapping dened by the contraction ω[(X) = iXω. We have that ω[ is C∞(M)-linear. Moreover, if ω is weakly (resp. strongly) non-degenerate, then ω[ is injective(resp. bijective).

Denition 1.28. Let M be a smooth manifold, a weak (resp. strong) symplectic form isa weakly (resp. strongly) non-degenerate 2-form ω ∈ Ω2(M) which is in addition closed.The pair (M,ω) is called a weak (resp. strong) symplectic manifold.


Theorem 1.29 (Darboux). Let ω be a 2-form over a nite-dimensional smooth manifoldM . Then, (M,ω) is a symplectic manifold if and only ifM has even dimension (dimM =2n) and there exist local coordinates (q1, . . . , qn, p1, . . . , pn) such that ω has locally the form

ω = dqi ∧ dpi.

Such coordinates are called Darboux or canonical coordinates.

Example 1.30 (T ∗Q as a symplectic manifold). Let Q be a smooth manifold of dimensionn and consider its cotangent bundle T ∗Q. We dene on T ∗Q a 1-form Θ ∈ Ω(T ∗Q) by

Θα(X) = α((TαπQ)(X)), X ∈ Tα(T ∗Q), α ∈ T ∗Q.

The 1-form Θ is known as the Liouville 1-form, or also as the canonical or tautological1-form. In adapted coordinates (qi, pi) of T ∗Q, Θ has the local expression

Θ = pi dqi.

We now dene on T ∗Q the canonical 2-form:

Ω = − dΘ.

From the local expression of Θ, we have that Ω is locally written as

Ω = dqi ∧ dpi

for the local coordinates (qi, pi) of T ∗Q. We thus infer that Ω is symplectic and hence itendows T ∗Q with a canonical symplectic structure, (T ∗Q,Ω).

1.4.1 The Gotay-Nester-Hinds algorithm

By denition, if (M,ω) is a strongly symplectic manifold (posibly of nite dimension),then the equation

iXω = α (1.7)

has always a unique solution X ∈ X(M), whatever the 1-form α ∈ T ∗M is (Proposition1.27). In the nite dimensional case and we suppose that dimM = 2n, the solution vectoreld X ∈ X(M) is locally given by

X = ω](α) = (ω[)−1(α) =2n∑i,j=1

ωijαj∂

∂xi, (1.8)

where (x1, . . . , x2n) are arbitrary local coordinates on M , ωij is the inverse coecientmatrix of ω, with ω =

∑1≤i<j≤2n ωij dxi ∧ dxj, and α =

∑2nj=1 αj dxj. If we instead

choose Darboux coordinates (q1, . . . , qn, p1, . . . , pn) for M and write

X = X i ∂

∂qi+Xi

∂

∂piand α = αi dq

i + αi dpi,

thenX i = αi and Xi = −αi. (1.9)

1.4. SYMPLECTIC GEOMETRY 9

This equations will appear again in later sections in slightly dierent ways.The aim of the Gotay-Nester-Hinds (GNH) algorithm (see reference [100, 101, 102])

is to study the equation (1.7) whenever the closed 2-form ω is weakly symplectic ordegenerate, that is, when ω is presymplectic. It manages to circumvent the degeneracyproblems that often appear in mechanics, even though it is totally geometric and maybe studied appart of any physical meaning. Equation (1.7) could not be solvable for apresymplectic form ω over the whole manifold M , but it could be at some points of M .The objective of the GNH algorithm is to nd a submanifold N of M such that equation(1.7) has solutions in N , more precisely, to nd the biggest submanifold N of M suchthat there exists a vector eld X ∈ X(N) that satises

ij∗Xω|N = α|N (1.10)

for a prescribed 1-form α ∈ Ω(M), where j is the inclusion j : N →M . The manifold Nwill, of course, depend on the 1-form α.

Remark 1.31. Even though they are quite similar, Equation (1.10) should not be confusedwith the following one

iX(j∗ω) = j∗α.

While the latter must be satised for any vector eld Y over N , that is

(j∗ω)(X, Y ) = (j∗α)(Y ), ∀Y ∈ X(N),

the former is more restrictive and must be satised for any vector eld Y along N , thatis

ω(j∗X, Y ) = α(Y ), ∀Y ∈ X(j).

Given a presymplectic 2-form ω over a manifold M , let α ∈ Ω(M) be any 1-form. Westart dening the subset M1 of points x of M such that α(x) is in the range of ω[(x),that is,

M1 :=x ∈M : α(x) ∈ ω[(TM)

.

The subset M1 needs not to be a manifold, fact that is imposed, being j1 : M1 →M theinclusion. The equation (1.7) restricted to M1,

iXω|M1 = α|M1 ,

is solvable, but this does not imply that X is a solution in the sense of equation (1.10).It could be possible that, at some point x ∈M1, the vector X(x) dont be tangent to M1,what will happen when α(x) dont be in the range of ω[(x) restricted to j1∗(TM1). Weare then obliged to dene a new submanifold j2 : M2 →M1 by

M2 :=x ∈M1 : α(x) ∈ ω[(j1∗(TM1))

.

As before, the solutions of the equation (1.7) restricted to M2,

iXω|M2 = α|M2 ,

may not be tangent to M2, therefore we require that α|M2 be in the range of ω[ restrictedto (j2 j1)∗(TM2).


We thus continue this process, dening a chain of further constraint submanifolds

. . . →Ml

jl→Ml−1 → . . . →M1

j1→M

as followsMl+1 :=

x ∈Ml : α(x) ∈ ω[((j1 · · · jl)∗(TMl))

. (1.11)

At each step, we must assume that the constraint set Ml is a smooth manifold (analternate algorithm for the case when the constraint sets are not smooth submanifoldsmay be found in [114]). In the end, the algorithm will stop when, for some k ≥ 0,Mk+1 = Mk. We then take N := Mk and j := jk · · · j1. Mainly, two dierent casesmay happen:

dimN = 0 : The Hamiltonian system (M,ω, α) has no dynamics. Furthermore, ifN = ∅, there are no solutions at all and (M,ω, α) does not accurately describe thedynamics of any system. On the contrary, if N 6= ∅, then N consists of (steady)isolated points.

dimN 6= 0 : (M,ω, α) describes a dynamical system restricted to N and we havecompletely consistent equations at motion on N of the form

(iXω − α)|N = 0.

1.5 Cosymplectic geometry

While symplectic geometry deals with even- dimensional spaces, cosymplectic geometryis the natural extension to study analog structures in odd-dimensional spaces. Throughthis section, V and M respectively denote a real vector space and a smooth manifold ofdimension 2n+ 1.

Denition 1.32. A cosymplectic vector space is a triple (V, ω, η) where V is an odd-dimensional real vector space (dimV = 2n+ 1), ω is a 2-form on V and η is a 1-form onV such that the exterior product ωn ∧ η is not null.

Proposition 1.33. Let V be an odd-dimensional real vector space (dimV = 2n + 1).Given a 2-form ω and a 1-form η on V , dene the morphism [ : V −→ V ∗ by

[(v) = ivω + η(v)η.

If (V, ω, η) is cosymplectic, then [ is an isomorphism. In that case, the vector R = [−1(η)is called the Reeb vector of the cosymplectic space (V, ω, η).

Note that the Reeb vector is characterized by the equations

iRω = 0, iRη = 1.

Example 1.34. Let V be a real vector space of dimension n. Let (ei)ni=1 be a basis of V and

let (εi)ni=1 be its dual counterpart. Dene ω =∑n

i=1 εi∧ ei, the canonical non-degenerate

2-form in V × V ∗ (see example 1.25). Let η be a non-zero covector in R. Then, withsome abuse of notation, (V × V ∗ × R, ω, η) is a cosymplectic vector space.

1.6. MULTISYMPLECTIC GEOMETRY 11

Denition 1.35. A cosymplectic manifold is a triple (M,ω, η) where M is an odd-dimensional smooth manifold (dimM = 2n + 1), ω is a closed 2-form on M and η bea closed 1-form on M such that (TxM,ωx, ηx) is a cosymplectic vector space for eachx ∈M .

Proposition 1.36. Let M be an odd-dimensional smooth manifold. Given a 2-form ωand a 1-form η on M , dene the map [ : X(M) −→ Ω(M) by

[(X) = iXω + η(X)η.

If (M,Ω, η) is cosymplectic then [ is an isomorphism of C∞(M)-modules. In that case,R = [−1(η) is known as the Reeb vector eld of the cosymplectic manifold (M,ω, η).

Again, note that the Reeb vector eld is characterized by the equations

iRω = 0, iRη = 1.

Proposition 1.37. Let M be an odd-dimensional smooth manifold (dimM = 2n + 1).Given a 2-form ω and a 1-form η on M , the triple (M,Ω, η) is a cosymplectic manifoldif and only if there exist local coordinates (q1, . . . , qn, p1, . . . , pn, t) such that ω and η havelocally the expression

Ω = dqi ∧ dpi, η = dt.

Such coordinates are called Darboux or canonical coordinates.

Example 1.38 (T ∗Q × R as a cosymplectic manifold). Let Q be a smooth manifold ofdimension n and consider its cotangent bundle T ∗Q. Let Ω be the canonical 2-form onT ∗Q (see example 1.30) and let η a volume form on R, for instance, η = dt. Then,(T ∗Q× R,Ω, η) is a cosymplectic manifold.

1.6 Multisymplectic geometry

For an introduction to multisymplectic geometry and its use within classical eld theory,the reader is refereed to [31, 32]. See also [124, 132].

Through this section, V and M respectively denote a real vector space and a smoothmanifold, both of nite dimension.

Denition 1.39. Amultisymplectic k-form on a real vector space V is a k-form ω ∈ ΛkV ∗

with trival kernel, kerω = 0, where the kernel is kerω := v ∈ V : ivω = 0. The pair(V, ω) is said to be a multisymplectic vector space.

A necessary condition to be satised by a multisymplectic k-form ω is that 1 < k ≤dimV . The non-degeneracy condition kerω = 0 is sometimes written as

ivω = 0⇔ v = 0.

Note also that a multisymplectic 2-form is a symplectic one.

Denition 1.40. Given a k-form ω ∈ ΛkV ∗, we dene the mappings

ω[j : ΛjV −→ Λk−jV ∗

v 7→ ivω

for any 1 ≤ j ≤ k.


If any of the linear maps ω[j is null, then ω must be the zero k-form (and conversely).Thus, ω[k−1 is surjective whenever ω is not zero. If ω is multisymplectic, then ω[1 isinjective.

Example 1.41. Given any real vector space V of dimension n, consider the product VV =V × ΛkV ∗, with 1 < k ≤ n. We dene the (k + 1)-form ΩV in VV by

ΩV ((v1, ω1), . . . , (vk+1, ωk+1)) :=k+1∑i=1

(−1)iωi(v1, . . . , vi, . . . , vk+1)

where (vi, ωi) ∈ VV for i = 1, . . . , k + 1 and where the hat symbol ˆ means that theunderlying vector is ommited. If (ei) is a basis for V , (εi) is the corresponding dual basisfor V ∗ and (ei1···ik = ei1 ∧ · · · ∧ eik) is the basis for ΛkV , where 1 ≤ i1 < · · · < ik ≤ k,then

ΩV =∑

1≤i1<···<ik≤k

−ei1···ik ∧ εi1 ∧ · · · ∧ εik .

It is easly seen from here that, when k = 1, we recover the symplectic 2-form given inthe example 1.25.

If E is a proper vector subspace of V , we denote by ΛkrV∗ the collection of k-forms in

V that are anihilated when r vectors of E are applied to it,

ΛkrV∗ =

α ∈ Λk

rV∗ : ivr · · · iv1α = 0, ∀v1, . . . , vr ∈ E

.

We then have that VrV = V × ΛkrV∗ equiped with the restriction of ΩV to it is a multi-

symplectic space. Note that, if E = 0, we then recover the whole VV .

Denition 1.42. A multisymplectic k-form on a smooth manifold M is a closed k-formω ∈ Ωk(M) such that (TxM,ωx) is a multisymplectic vector space for each x ∈ M . Thepair (M,ω) is said to be a multisymplectic manifold.

Example 1.43. Given a smooth manifold of dimension n, consider the ber bundle ΛkMof k-forms. We dene on ΛkM the k-form Θ ∈ Ωk(ΛkM) by

Θα(X1, . . . , Xk) = α((TαπkM)(X1), . . . , (Tαπ

kM)(Xk)), Xi ∈ Tα(ΛkM), α ∈ ΛkM,

where πkM : ΛkM → M is the canonical projection. The k-form Θ is known as theLiouville k-form, or also as the canonical or tautological k-form. In adapted coordinates(qi, pi1···ik) of ΛkM , Θ has the local expression

Θ =∑

1≤i1<···<ik≤k

pi1···ik dqi1 ∧ · · · ∧ dqik .

We now dene on ΛkM the canonical (k + 1)-form:

Ω = − dΘ.

From the local expression of Θ, we have that Ω is locally written as

Ω =∑

1≤i1<···<ik≤k

− dpi1···ik ∧ dqi1 ∧ · · · ∧ dqik

1.6. MULTISYMPLECTIC GEOMETRY 13

for the local coordinates (qi, pi1···ik) of ΛkM . We thus infer that Ω is multisymplectic andhence it endows ΛkM with a canonical multisymplectic structure, (ΛkM,Ω). It is easilyseen from here that, when k = 1, we recover the canonical symplectic 2-form given in theexample 1.30.

If M bers over a manifold N , π : M → N , we denote by ΛkrM the collection of

k-forms over M that are anihilated when r π-vertical vectors are applied to it,

ΛkrM =

α ∈ ΛkM : ivr · · · iv1α = 0, ∀v1, . . . , vr ∈ V π

.

We then have that ΛkrM equiped with the restriction of Ω to it is a multisymplectic space.

Nota that, if N = M , we then recover the whole ΛkM .

Chapter 2

Classical Mechanics

2.1 The Lagrangian formalism for autonomous systems

The Lagrangian formulation of mechanics is set (for simplicity) in a nite dimensionalmanifold Q (the innite dimensional case is depicted in [122]), the conguration space,whose tangent, TQ, describes the states position plus velocity of the system understudy. Local coordinates (qi) on Q induce ber coordinates (qi, vi) on TQ, such that atangent vector v ∈ TqQ at some point q ∈ Q is written as

v = vi∂

∂qi= v1 ∂

∂q1+ · · ·+ vn

∂

∂qn.

One introduces the Lagrangian of the system, a smooth function L : TQ −→ R, whichis in some sense the density cost of a motion in the system. Typically, the Lagrangian isthe kinetic energy minus the potential energy of the system,

L(qi, vi) =1

2mgijv

ivj − U(qi) (L(qi, vi) =1

2mg(vq, vq)− U(q)),

where gij = gij(q) is a given metric tensor and m the mass of the particle in motion.We seek for curves that describe the motion of a particle in our system. It is well

known that the trajectories of the system are obtained from a variational procedure. Wewill thus consider twice dierentiable curves c : [t0, t1] → Q joining two xed points q0

and q1 in Q. The set of such curves is

C2([t0, t1], Q, q0, q1) =c ∈ C2([t0, t1], Q) : c(ti) = qi, i = 0, 1

,

or C2(q0, q1) for short. Given c ∈ C2(q0, q1), denote by c(t) its lift to TQ, that is, thecurve in TQ that describes the position and velocity of a particle following the originalcurve (see Denition 1.20). Formally, c is the vector eld over c such that

c(t)(f) =d

dt(f c)(t),

for any smooth function f ∈ C∞(Q). If (qi, vi) are adapted coordinates in TQ, then

c(t) = (qi(t), vi(t)),

where one regards vi = dqi/ dt as the velocity of a particle moving along c(t).If (qi, vi) are adapted coordinates on TQ, locally

c(t) = (ci(t), ci(t)),

where ci(t) = (qi c)(t) = (qi c)(t) and ci(t) = vi c)(t) = ( dci/ dt)(t) .

15

16 CHAPTER 2. CLASSICAL MECHANICS

2.1.1 Variational approach

Denition 2.1. Given a Lagrangian function L : TQ −→ R, two xed points q0, q1 ∈ Qand a xed time interval [t0, t1], the associated integral action is the real valued map Adened on C2([t0, t1], Q, q0, q1) given by

AL(c) =

∫ t1

t0

L(c(t)) dt =

∫ t1

t0

L(qi(t), vi(t)) dt. (2.1)

Since we look for a variational approach of the problem, we must describe how theintegral action AL changes under small perturbations of c and what these perturbationsare. One shows that C2(q0, q1) may be endowed with an innite-dimensional smoothmanifold structure, see [21]. In fact,

TcC2(q0, q1) =δc ∈ C1([t0, t1], TQ) : τQ δc ≡ c, δc(ti) = 0, i = 0, 1

.

Denition 2.2. Let c ∈ C2([t0, t1], Q, q0, q1), a variation of c is a curve cs in C2(q0, q1),dened for a small interval [−ε, ε], such that c0 ≡ c. An innitesimal variation of c is avector eld δc over c that vanishes at the end points, δc(ti) = 0, for i = 0, 1.

With this denition, the tangent space TcC2(q0, q1) at a curve c ∈ C2(q0, q1) is the setof innitesimal variations δc of c, which are induced by variations cs of c. More precisely,

δc(t) =dcs(t)

ds

∣∣∣s=0

,

where t ∈ [t0, t1] is xed.

Denition 2.3. Let F : C2(q0, q1) −→ R be a functional of class C1. A critical point ofF is a point c ∈ C2(q0, q1) such that

d(F cs)ds

∣∣∣s=0

= 0,

for any variation cs of c.

Equivalently, one could say that c is a critical point of a functional F ∈ C1(C2(q0, q1))if and only if

δF(c) · δc = 0,

for any innitesimal variation δc of c, which is classically written as

δF(c) = 0 orδF(c)

δc= 0.

We are now in position to formulate one of the main results in Classical Mechanics, thevariational principle of Hamilton, which states that the dynamics of our physical systemis determined from the variational problem related to the integral action AL.

Statement 2.4 (Hamilton's principle). The motion of a particle in the Lagrangian sys-tem (Q,L) is a critical point of the action functional AL, that is, a curve c ∈ C2(q0, q1)such that δAL(c) = 0.

2.1. THE LAGRANGIAN FORMALISM FOR AUTONOMOUS SYSTEMS 17

An easy calculation help us to write this statement in terms of the Lagrangian, ob-taining the well known Euler-Lagrange equations.

Theorem 2.5 (The Euler-Lagrange equations). Consider a given Lagrangian system(Q,L), where L ∈ C2(TQ). A twice dierentiable curve c : [t0, t1] −→ Q joining twopoints q0, q1 ∈ Q is a motion of (Q,L) if and only if the lift c of c to TQ satises thedierential equations:

∂L

∂qi c− d

dt

(∂L

∂vi c)

= 0, (2.2)

where (qi, vi) are adapted coordinates in a neighborhood of c.

Proof. Given a curve c ∈ C2(q0, q1), let δc be an innitesimal variation of c tangent to avariation cs of c. By denition and dierentiating under the integral sign, we have that

δAL(c) · δc =d(AL cs)

ds

∣∣∣s=0

=d

ds

[∫ t1

t0

L(cs(t)) dt

] ∣∣∣s=0

=

∫ t1

t0

d

ds

[L(cis(t), c

is(t))

] ∣∣∣s=0

dt

=

∫ t1

t0

[∂L

∂qi· δci(t) +

∂L

∂vi· δci(t)

]dt,

provided that ε is small enough such that the image of cs : [−ε, ε] × [t0, t1] −→ TQis covered by a single chart with adapted coordinates (qi, vi). Integrating by parts thesecond term and taking into account that δc vanishes at t0 and t1, we obtain

δAL(c) · δc =

∫ t1

t0

[∂L

∂qi− d

dt

(∂L

∂vi

)]· δci(t) dt.

Now, let us suppose that c is a motion of the system. Then, δAL(c) · δc = 0 for everyinnitesimal variation δc, which holds if and only if

∂L

∂qi− d

dt

∂L

∂vi= 0.

2.1.2 Geometric formulation

Denition 2.6. The Poincaré-Cartan 1-form is the pull-back of the dierential of theLagrangian function by the vertical endomorphism S, that is, the form

ΘL = S∗( dL). (2.3)

The Poincaré-Cartan 2-form is then given by

ΩL = − dΘL. (2.4)


In adapted coordinates, the Poincaré-Cartan 1-form reads

ΘL =∂L

∂vidqi (2.5)

and the Poincaré-Cartan 2-form

ΩL =∂2L

∂vi∂qjdqi ∧ dqj +

∂2L

∂vi∂vjdqi ∧ dvj. (2.6)

By denition, the Poincaré-Cartan 2-form is exact, and hence closed, but in general needsnot to be non-degenerate.

Proposition 2.7. The Poincaré-Cartan 2-form is non-degenerate if and only if the La-grangian function is regular, that is when the Hessian matrix(

∂2L

∂vi∂vj

)(2.7)

is invertible.

Denition 2.8. The Lagrangian energy is the smooth function EL ∈ C∞(TQ) denedby

EL = ∆L− L, (2.8)

where ∆ denotes the Liouville vector eld given in Denition 1.18.

Denition 2.9. Any vector eld X ∈ X(TQ) that satises the equation of motion

iXΩL = dEL (2.9)

is called a Lagrangian vector eld.

Theorem 2.10. If the Lagrangian function L is regular, then there exists a unique vectoreld X ∈ X(TQ) which is solution of the equation of motion. The Lagrangian vector eldX is a second order dierential equation and its base integral curves are solutions of theEuler-Lagrange equations (2.2).

Proof. The existence and uniqueness of a Lagrangian vector eld comes out from the factthat ΩL is non-degenerate when L is regular, hence ΩL is symplectic and Proposition 1.27applies. Let X be a generic vector eld on TQ whose local expression is

X = qi∂

∂qi+ vi

∂

∂vi

for adapted coordinates (qi, vi) of TQ and let suppose that X satises the equation (2.9).The contraction of the Poincaré-Cartan 2-form ΩL with X is

iXΩL =

(qj

∂2L

∂vj∂qi− qj ∂2L

∂vi∂qj− vj ∂2L

∂vj∂vi

)dqi + qj

∂2L

∂vj∂vidvi,

and the dierential of the Lagrangian energy EL is

dEL =

(vj

∂2L

∂vj∂qi− ∂L

∂qi

)dqi + vj

∂2L

∂vj∂vidvi.

2.2. THE HAMILTONIAM FORMALISM FOR AUTONOMOUS SYSTEMS 19

Equating coecients, we have on the one hand that

qj∂2L

∂vj∂vi= vj

∂2L

∂vj∂vi.

Thus, if L is regular, qj = vj, which proves that X is second order. On the other hand,provided that L is regular, we have that

∂L

∂qi− vj ∂2L

∂qj∂vi− vj ∂2L

∂vj∂vi= 0.

Let c : I −→ Q be a base integral curve of the Lagrangian vector eld X, then qi = ci =dc/ dt and vi = ci = d2c/ dt2. Substituting this in the previous equation and denotingc = (ci, ci) the lift of c to TQ, we obtain

0 =∂L

∂qi c− dcj

dt

∂2L

∂qj∂vi c− dcj

dt

∂2L

∂vj∂vi c =

∂L

∂qi c− d

dt

(∂L

∂vi c),

which are precisely the Euler-Lagrange equations (2.2).

2.2 The Hamiltoniam formalism for autonomous sys-

tems

For a more extended description of the Hamiltonian formalism, please refer to [1, 64, 122].As for the Lagrangian formalism, the Hamiltonian formulation of mechanics is set

in a nite dimensional manifold Q, the conguration space, but in contrast, the statesposition plus momentum of the system under study are described by the cotangentbundle T ∗Q of Q. Local coordinates (qi) on Q induce ber coordinates (qi, pi) on T ∗Q,such that a 1-form α ∈ T ∗qQ at some point q ∈ Q is written as

α = pi dqi = p1 dq1 + · · ·+ pn dqn.

One introduces the Hamiltonian of the system, a smooth function H : T ∗Q −→ R, whichis in some sense the is the total energy density of the system being described. Typically,the Hamiltonian is the kinetic energy plus the potential energy of the system,

H(qi, pi) = K(pi) + U(qi) (= T (pi) + V (qi)).

Denition 2.11. Given a Hamiltonian function H : T ∗Q −→ R, the Hamiltonian vectoreld with energy function H is the unique vector eld X ∈ X(M) such that

iXHΩ = dH,

where Ω is the canonical symplectic form of T ∗Q.

Theorem 2.12 (Hamilton's equations). A dierentiable curve c : I −→ T ∗Q is anintegral curve of XH if and only if the Hamilton's equations hold:

qi =∂H

∂piand pi =

∂H

∂qi,

where c(t) = (qi(t), pi(t)).

Proposition 2.13. Given an integral curve c(t) of XH , we have that H(c(t)) is constant.

Proposition 2.14. Let Ft ∈ Diff(M) be the ow of XH , then F ∗t ω = ω, for each t, i.e.Ft is a family of symplectomorphisms.


2.3 The Legendre transformation

The Legendre transformation is the master key that relates the Lagrangian and the Hamil-tonian formalisms. Although the technique is usually used to go from the Lagrangian sideto the Hamiltonian one, it can be restated to pass from the latter to the former.

Denition 2.15. Given a Lagrangian function L : TQ −→ R, the Legendre transforma-tion associated to L is the bered mapping legL : TQ −→ T ∗Q given by

〈legL(v), w〉 :=d

dεL(v + ε · w)

∣∣∣ε=0

.

If (qi, vi) and (qi, pi) denote ber coordinates on TQ and T ∗Q respectively, we thenhave

legL(qi, vi) =

(qi, pi =

∂L

∂vi

).

Proposition 2.16. If L is regular, then legL : TQ −→ T ∗Q is a local dieomorphism.

Denition 2.17. A Lagrangian function L : TQ −→ R is said to be hyper-regularwhenever legL is a global dieomorphism.

Theorem 2.18. Given a Lagrangian function L : TQ −→ R, we have that

ΘL = leg∗L Θ and ΩL = leg∗L Ω.

Moreover, if L is hyper-regular and we dene the Hamiltonian

H := EL leg−1L = vi

∂L

∂vi− L,

then the Lagrangian vector eld XL and the Hamiltonian vector eld XH are legL-related,i.e. XH = (legL)∗XL.

2.4 The Tulczyjew's triple

In [147, 148], W. Tulczyjew introduced a purely geometric construction based on a tripleof tangent and cotangent bundles in which the theory of classical mechanics ts perfectly.While its extension to higher-order mechanics has been completely achieved (see [30, 45,48]), there have been some attempts to reproduce it for eld theory but with only partialsuccess (for instance, [52, 109]).

Before we give the full picture, we start with two basic denitions.The canonical involution of TTQ is the smooth map κQ : TTQ −→ TTQ given by

κM

(d

ds

(d

dtχ(s, t)

∣∣∣t=0

) ∣∣∣s=0

):=

d

ds

(d

dtχ(s, t)

∣∣∣t=0

) ∣∣∣s=0

,

where χ : R2 −→ Q and χ(s, t) := χ(t, s). Note that ddtχ(s, t)|t=0 : R −→ TQ.

The tangent pairing between TT ∗Q and TTQ is the bered map 〈·, ·〉T : TT ∗Q ×QTTQ −→ R given by⟨

d

dtγ(t)

∣∣∣t=0,

d

dtδ(t)

∣∣∣t=0

⟩T:=

d

dt〈γ(t), δ(t)〉T

∣∣∣t=0,

where γ : R −→ T ∗Q and δ : R −→ TQ are such that πQ γ ≡ τQ δ.

2.4. THE TULCZYJEW'S TRIPLE 21

Denition 2.19. The Tulczyjew's isomorphism is the map α : TT ∗Q −→ T ∗TQ givenby

〈α(V ),W 〉 := 〈V, κQ(W )〉T , V ∈ TT ∗Q, W ∈ TTQ.

In coordinates,α(qi, pi, qi, pi) = (qi, qi, pi, pi).

Denition 2.20. Dene the map β : TT ∗Q −→ T ∗T ∗Q by

β(V ) := iV Ω, V ∈ TT ∗Q,

where Ω is the canonical symplectic form of T ∗Q.

In coordinates,β(qi, pi, qi, pi) = (qi, qi, pi, pi).

By means of α and β, TT ∗Q may be endowed with two (a priori) dierent symplecticstructures: Let ΩTQ and ΩT ∗Q be the canonical symplectic forms of T ∗TQ and T ∗T ∗Q(as cotangent bundles), respectively. Then, both of Ωα = α∗ΩTQ and Ωβ = β∗ΩT ∗Q denesymplectic structures on TT ∗Q which turn out to be the same; more precisely, Ωα = −Ωβ.Moreover, there is a third canonical symplectic structure on TT ∗Q which comes from thecomplete lift of the canonical symplectic form ΩQ of Q to TT ∗Q, which we denote Ω

(1)Q ,

and which coincides with the previous ones; more precisely, Ω(1)Q = Ωα. In coordinates,

Θα = α∗ΘTQ = p dq + p dq and Θβ = β∗ΘT ∗Q = −p dq + q dp,

where ΘTQ and ΘT ∗Q are the Liouville 1-forms on TQ and T ∗Q, respectively.

Theorem 2.21. Given a Hamiltonian function H : T ∗Q −→ R, consider the associatedHamiltonian vector eld XH ∈ X(T ∗Q). The following holds,

1. The image of XH is a Lagrangian submanifold SXH of (TT ∗Q,Ωβ).

2. The image of dH is a Lagrangian submanifold S dH of (T ∗T ∗Q,ΩT ∗Q).

3. The isomorphism β maps one into another, i.e. β(SXH ) = SdH .

Lemma 2.22. Given a Lagrangian function L : TQ −→ R, then the image of dL is aLagrangian submanifold S dL of (T ∗TQ,ΩTQ).

Proposition 2.23. Given an hyper-regular Lagrangian function L : TQ −→ R, considerthe associated Hamiltonian H = EL leg−1

L . We have that α−1(SdL) = SXH = β−1(S dH)

T ∗TQ

πTQ

88888888888888TT ∗Qαoo β //

TπQ

τT∗Q

99999999999999T ∗T ∗Q

πT∗Q

TQlegL //

dL

NN

T ∗Q

dH

PP

Chapter 3

Classical Field Theory

The main reference for this chapter is the book by Saunders [139], although it does notcover all the sections (references will be provided when necessary). Besides, other basicreferences are [21, 91, 97, 109, 61, 136].

3.1 Jet bundles

Through this section, (E, π,M) denotes a ber bundle whose base space M is a smoothmanifold of dimension m, and whose bers have dimension n, thus E is (m + n)-dimensional. Adapted coordinate systems in E will be of the form (xi, uα), where (xi) isa local coordinate system in M and (uα) denotes ber coordinates.

Denition 3.1. Given a point x ∈ M , two local sections φ, ψ ∈ Γxπ are 1-equivalent atx if their value coincide at x, φ(x) = ψ(x), as well as their tangent maps, Txφ = Txψ.This denes an equivalence relation in Γxπ. The equivalence class containing φ is calledthe rst order jet of φ at x and is denoted j1

xφ.

An alternative denition of the previous equivalence relation would be in terms ofpartial derivatives. Let (xi, uα) be a system of adapted local coordinates around φ(x), ψwill be 1-equivalent to φ at x if and only if

φα(x) = ψα(x) and∂φα

∂xi

∣∣∣∣x

=∂ψα

∂xi

∣∣∣∣x

. (3.1)

Denition 3.2. The rst order jet manifold of π, denoted J1π, is the whole collectionof rst order jets of arbitrary local sections of π, that is,

J1π :=j1xφ : x ∈M, φ ∈ Γxπ

.

The functions given byπ1 : J1π −→ M

j1xφ 7−→ x

(3.2)

andπ1,0 : J1π −→ E

j1xφ 7−→ φ(x)

(3.3)

are called the source projection and the target projection respectively, and are smoothsurjective submersions.

23

24 CHAPTER 3. CLASSICAL FIELD THEORY

Proposition 3.3. The rst jet manifold of π, J1π, may be endowed with a structureof smooth manifold. A system of adapted coordinates (xi, uα) on E induces a systemof coordinates (xi, uα, uαi ) on J1π such that xi(j1

xφ) = xi(x), uα(j1xφ) = uα(φ(x)) and

uαi (j1xφ) = ∂φα

∂xi

∣∣x.

In the induced local coordinates (xi, uα, uαi ), the source and the target projections arewritten

π1(xi, uα, uαi ) = (xi) and π1,0(xi, uα, uαi ) = (xi, uα). (3.4)

From here, it is clear that π1 and π1,0 are certainly projections (surjective submersions)over M and E, respectively. Therefore, (J1π, π1,M) and (J1π, π1,0, E) are ber bundles.If we consider a change of coordinates (xi, uα) 7→ (yj(xi), vβ(xi, uα)) in E, it inducesa change of coordinates (xi, uα, uαi ) 7→ (yj(xi), vβ(xi, uα), vβj (xi, uα, uαi )) in J1π. In thiscase, the velocities transform by the following rule:

vβj =

(∂vβ

∂xi+∂vβ

∂uαuαi

)∂xi

∂yj. (3.5)

Note that the change of coordinates is not linear, like in the tangent bundle, but ane.

Proposition 3.4. The rst jet manifold of π, J1π, together with the target projection,π1,0, is an ane bundle over E. The ber in J1π over a point u ∈ Ex, J1

uπ, is dieomor-phic to the ane space

A ∈ Lin(TxM,TuE) : Tuπ A = IdTxM .The underlying vector bundle has typical ber

A ∈ Lin(TxM,TuE) : Tuπ A = 0 = Lin(TxM,Vu π).

Moreover, the induced coordinate systems (xi, uα, uαi ) are adapted to the ane bundlestructure.

Formally, the associated vector bundle to J1π is the bundle over E whose total spaceis the tensor product T ∗M ⊗E V π, that is, the bundle

(T ∗M ⊗E V π, (τE|V π) pr 2, E).

Let j1xφ ∈ J1π and consider a typical element A ∈ T ∗xM ⊗ Vφ(x) π, the action of A on

j1xφ is the 1-jet j1

xψ = j1xφ + A such that ψ(x) = φ(x) and Txψ = Txφ + A. In adapted

coordinates,uαi (j1

xφ+ A) = uαi (j1xφ) + Aαi ,

where

A = Aαi dxi ⊗ ∂

∂uα.

Despite (J1π, π1,0, E) is ane, if we consider a preferred global section and see it asthe zero section, one could thought of J1π as a vector bundle. Obviously, in general,there is no such preferred global section. But, when E is trivial, there it is. Suppose thatE = M × F . For each u ∈ E we dene the constant section φu ∈ Γπ by

φu(x) := (x, pr 2(u)).

We then dene the zero section z ∈ Γπ1,0 by

z(u) := j1xφu = (xi, uα, uαi = 0).

In the particular case where π is the bundle (R× F, pr 1,R), J1π turns to be isomorphicto R× TF .

3.1. JET BUNDLES 25

3.1.1 Prolongations, lifts and contact

Denition 3.5. Let φ ∈ Γπ be a (local) section, its rst prolongation is the (local)section of π1,0 given by

(j1φ)(x) := j1xφ,

for every x ∈ M . An arbitrary (local) section σ of π1 is said to be holonomic if it is therst prolongation of a (local) section φ ∈ Γπ, that is, if σ = j1φ.

Denition 3.6. Let f : E → F be a morphism between two ber bundles (E, π,M) and(F, ρ,N), such that the induced function on the base, f : M → N , is a dieomorphism.The rst prolongation of f is the map j1f : J1π → J1ρ given by

(j1f)(j1xφ) := j1

f(x)φf , ∀j1

xφ ∈ J1π,

where φf := f φ f−1.

J1πj1f //

π1,0

J1ρ

ρ1,0

E

f //

π

F

ρ

M

f //

φ

HHj1φ

FF

N

φf

VVj1φf

XX

Note that the rst prolongation j1f of a morphism f is both, a morphism between(J1π, π1,0, E) and (J1ρ, ρ1,0, F ), and a morphism between (J1π, π1,M) and (J1ρ, ρ1, N).In each case, the induced functions between the base spaces are f and f , respectively.

If (xi, uα, uαi ) and (yj, vβ, vβj ) denote adapted coordinates in J1π and J1ρ, respectively,then we have

yj(j1f) = f j, vβ(j1f) = fβ and vβj (j1f) =

(∂fβ

∂xi+ uαi

∂fβ

∂uα

)· ∂f

−i

∂yj.

The expression between brackets in the last equation is called the total derivative of fβ

with respect to xi. We will come back to it later.

Denition 3.7. Let φ : M → E be a section of π, x ∈ M and u = φ(x). The verticaldierential of the section φ at the point u ∈ E is the map

dvuφ : TuE −→ Vu π

v 7−→ v − Tu(φ π)(v)

Namely, dvuφ := Idu−Tu(φ π).


Notice that the image of dvuφ is certainly in Vu π since Tuπ dv

u = 0 and that, in fact,dvuφ depends only on j1

xφ. In adapted local coordinates (xi, uα) of E,

dvuφ =

(duα − ∂φα

∂xidxi)⊗ ∂

∂uα. (3.6)

Denition 3.8. The canonical structure form of J1π is the 1-form θ on J1π with valuesin V π dened by

θj1xφ(V ) := ( dvφ(x)φ)(Tj1xφπ1(V )), V ∈ Tj1xφJ

1π, (3.7)

where φ is any representative of j1xφ ∈ J1π. The contraction of the covectors in V∗ π with

θ denes a distribution in T ∗J1π. This distribution is called the contact module or theCartan codistribution (of order 1) and it is denoted C1. Its elements are contact forms.The annihilator of C1 is the Cartan distribution (of order 1).

This is the approach taken by Echevaría-Enríquez et al. in [71]. In Saunders' ter-minology (see [139], pages 136137), θ is one of the elements that conform the contactstructure of π1, which is given by a natural decomposition in π∗1,0(τE), what is out of ourscope.

Note that the expression (3.7) does not depend on the representative φ of j1xφ, hence

it is well dened. In adapted local coordinates (xi, uα, uαi ) of J1π,

θ =(

duα − uαi dxi)⊗ ∂

∂uα. (3.8)

In fact, the contact forms duα − uαi dxi ∈ C1 are a base of the contact module.

Proposition 3.9. Let (xi, uα, uαi ) be adapted coordinates on J1π, a basis of the Cartancodistribution is given by the coordinate or local contact forms

θα = duα − uαi dxi. (3.9)

Proposition 3.10. The canonical structure form θ ∈ Γ(T ∗J1π⊗J1π V π) and the contactforms ω ∈ C1 are pulled back to zero by the rst prolongation j1φ of any section φ of π.Moreover, this characterizes the module of contact forms, i.e.

ω ∈ C1 ⇔ (j1φ)∗ω = 0, ∀φ ∈ Γπ. (3.10)

A complementary or dual result to the previous one is the following.

Proposition 3.11. Let σ ∈ Γπ1 be a (local) section. The following statements are equiv-alent:

1. σ is holonomic.

2. σ pulls back to zero any contact form, that is

σ∗ω = 0, ∀ω ∈ C1. (3.11)

3. σ pulls back to zero the canonical structure form, that is

σ∗θ = 0. (3.12)

3.1. JET BUNDLES 27

Notice that the contact forms are π1,0-basic, which is clear from the coordinate ex-pression (3.9). Though, therefore they may be thought as forms along π1,0 rather thanon J1π. In this sense are dened total derivatives.

Denition 3.12. A total derivative is a vector eld ξ along π1,0 which is annihilatedby the Cartan codistribution (considered now as forms along π1,0). Given a system ofadapted coordinates (xi, uα, uαi ) in J1π, the local vector elds dened along π1,0 by

d

dxi=

∂

∂xi+ uαi

∂

∂uα(3.13)

are called coordinate total derivatives.

It is immediate to check that coordinate total derivatives are total derivatives, infact they dene a basis of such vector elds. Under a change of coordinates, (xi, uα, uαi )to (yj, vβ, vβj ), a coordinate total derivative transforms linearly by the Jacobian of theunderlying change of coordinates:

d

dyj=∂xi

∂yjd

dxi.

If ξ ∈ X(π1,0) is a total derivative with the coordinate representations

ξ = ξid

dxi= ξj

d

dyj,

where the coecients ξi and ξj are functions on J1π, then

ξi = ξj∂xi

∂yj.

Denition 3.13. The total lift of a vector eld ξ = ξi∂i on M is the unique totalderivative that projects on ξ itself, that is, the vector eld ξ ∈ X(J1π) locally given by

ξ(j1xφ) = ξi(x)

d

dxi

∣∣∣j1xφ. (3.14)

Note that the total lift of the coordinate partial derivatives in M are precisely thecoordinate total derivatives.

Now, consider the action of total derivatives on smooth functions over E. If f ∈C∞(E), the action of d/ dxi on it yields a function df/ dxi ∈ C∞(J1π). In particular,the action of d/ dxi on the coordinate function uα ∈ C∞(E), gives as expected

duα

dxi= uαi ∈ C∞(J1π).

Another interesting fact is how total derivatives and jets are related. Let f ∈ C∞(E),φ ∈ Γπ and ξ ∈ X(M), we have

ξ(f φ) = ξ(f) j1φ, (3.15)

and in coordinates∂(f φ)

∂xi=

df

dxi j1φ. (3.16)


Finally, note that coordinate total derivatives and ordinary partial derivates do not nece-sarilly conmute:

∂

∂xidf

dxj=

d

dxj∂f

∂xi,

∂

∂uαdf

dxj=

d

dxj∂f

∂uαbut

∂

∂uαi

df

dxj=

d

dxj∂f

∂uαi+ δji

∂f

∂uα,

where f ∈ C∞(E).

Denition 3.14. Given a vector eld ξ on E, its rst lift (or rst jet) is the unique vectoreld ξ(1) on J1π that is projectable to ξ by π1,0 and preserves the Cartan codistributionwith respect to the Lie derivative, i.e. Lξ(1)ω ∈ C1 for any ω ∈ C1.

Proposition 3.15. Let ξ be a vector eld on E. If ξ has the local expression

ξ = ξi∂

∂xi+ ξα

∂

∂uα(3.17)

in adapted coordinates (xi, uα) on E, then its rst lift ξ(1) has the form

ξ(1) = ξi∂

∂xi+ ξα

∂

∂uα+

(dξα

dxi− uαj

dξj

dxi

)∂

∂uαi(3.18)

for the induced coordinates (xi, uα, uαi ) on J1π.

Originally, the rst lift is dened for π-projectable vector elds on E. The rst lift ofsuch vector eld ξ is the innitesimal generator of the rst lift of the ow of ξ. Denition3.14 is a characterization of this property and it is generalized for any kind of vector eldson E (see [71]).

Proposition 3.16. Let ψε be the ow of a given π-projectable vector eld ξ over E.Then, the ow of ξ(1) is the rst prolongation of ψε, j1ψε.

3.1.2 The vertical endomorphisms

Denition 3.17. Given a 1-jet j1xφ ∈ J1π, let A ∈ T ∗xM ⊗Vφ(x) π. The vertical lift of A

at j1xφ is the tangent vector Av

j1xφ∈ Tj1xφ(J1π) given by

Avj1xφ

(f) =d

dtf(j1

xφ+ tA)∣∣t=0, ∀f ∈ C∞(J1

φ(x)π). (3.19)

By the very denition of vertical lift, given a smooth function f ∈ C∞(E),

(Tj1xφπ1,0)(Avj1xφ

)(f) = Avj1xφ

(f π1,0)

=d

dt(f π1,0)(j1

xφ+ tA)∣∣t=0

=d

dtf(φ(x))

∣∣t=0

= 0.

Thus, the vertical lift takes values into the vertical ber bundle V π1,0 ⊂ TJ1π. Indeed,it is a morphism of vector bundles over the identity of J1π,

(·)v : T ∗M ⊗J1π V π −→ V π1,0.

3.1. JET BUNDLES 29

Note that, this time, the tensor product is taken over J1π and not over E. Note also thatfor each j1

xφ ∈ J1π, the vertical lift at j1xφ,

(·)vj1xφ

: T ∗xM ⊗ Vφ(x) π −→ Vj1xφ π1,0 ⊂ Tj1xφJ1π,

is a linear isomorphism. In adapted local coordinates (xi, uα, uαi ), if A = Aαi dxi|x ⊗∂/∂uα|φ(x), then

Avj1xφ

= Aαi∂

∂uαi

∣∣∣j1xφ

and (·)v = duα ⊗ ∂

∂xi⊗ ∂

∂uαi. (3.20)

Denition 3.18. Let η ∈ ΛmM be an arbitrary m-form on M . The vertical endomor-phism associated to η is the vector valued m-form Sη : (TJ1π)m −→ TJ1π that gives

Sη(V1, . . . , Vm) :=m∑i=1

ηi ⊗ [(Tj1xφπ1,0)(Vi)− (Tj1xφ(φ π1))(Vi)]

v, (3.21)

for any m tangent vectors V1, . . . , Vm ∈ Tj1xφJ1π, and where ηi is the contraction

ηi := (−1)m−iηx(V1, . . . , Vi, . . . , Vm)

with the hatted factor omitted.

Denition 3.19. The (canonical) vertical endomorphism S arises from the natural con-traction between the factors in V π of the structure canonical form θ and the factors inV∗ π of the vertical lift (·)v; that is

S := 〈θ, (·)v〉 ∈ Γ(T ∗J1π ⊗J1π TM ⊗J1π V π1,0). (3.22)

In adapted coordinates (xi, uα, uαi ) of J1π, the vertical endomorphisms have the localexpressions

Sη = ( duα − uαj dxj) ∧ dm−1xi ⊗∂

∂uαi= θα ∧ dm−1xi ⊗

∂

∂uαi(3.23)

and

S = ( duα − uαj dxj)⊗ ∂

∂xi⊗ ∂

∂uαi= θα ⊗ ∂

∂xi⊗ ∂

∂uαi, (3.24)

where θα = duα − uαj dxj are the local contact forms and dm−1xi = i∂/∂xi dmx.

3.1.3 Partial Dierential Equations

Lemma 3.20. If N is an open submanifold of M , then J1(πN) ' π−11 (N), where πN :=

π|π−11 (N).

Denition 3.21. A rst-order dierential equation on π is a closed embedded subman-ifold P of the rst jet manifold J1π. A solution of P is a local section φ ∈ ΓNπ, whereN is an open submanifold of M , which satises j1

xφ ∈ P for every x ∈ N . A rst-orderdierential equation P is said to be integrable at z ∈ P if there is a solution φ of P(around some neighborhood N of π1(z)) such that z = j1

π1(z)φ. A rst-order dierentialequation P is said to be integrable in a subset P ′ ⊂ P if it is integrable at each z ∈ P ′. Arst-order dierential equation P is said to be integrable if it is integrable at each z ∈ P .


If l is the codimension of P (dim J1π − dimP), there locally exist submersions Ψ :J1π → Rl for whom P is the zero level set. Written in local coordinates, P is given bythe set of points that satisfy

Ψµ(xi, uα, uαi ) = 0, µ = 1, . . . , l.

Thus, rst-order dierential equations are a geometric interpretation of the usual rst-order partial dierential equations. Under certain conditions, one could solve the previousequation for some of the velocities uαi making them to depend on the other variables(then π1,0|P : P → E would be a submersion). For simplicity, if n = 1 and 1 < l < m,rearranging conveniently the base variables, the previous equation could be equivalent tothe following expression

um−l+µ = φµ(xi, u, u1, . . . , um−l), µ = 1, . . . , l.

In the general case, if one projects P to E by π1,0, he would obtain a subset P(0,0)

of E, let us assume it is a smooth submanifold, which is not necessarily the whole of E.In such a case, it means that we are dealing with some constraint on the total space Eitself. An integral holonomic section j1φ of P will be such that the image of φ belongs toP(0,0) and the image of j1φ belongs to P(1,1) := J1P(0,0) ∩ P . The submanifold P(1,1) ofJ1π introduces new constraints that a solution of P must satisfy, moreover it representsthe rst step of the extension to jet bundles of the algorithm to extract the integral partof a dierential equation in a tangent bundle, which was presented by Mendella et al. in[127] (see also [126, 128]). The general algorithm will be given in Section 4.1.3.

Example 3.22. Given the ber bundle pr 1 : R2 → R2 × R3 with global coordinates(x, y, u, v, w), consider the constraint submanifold of J1 pr 1

P = (x, y, u, v, w, ux, uy, vx, vy, wx, wy) ∈ J1 pr 1 :

u = 0, v = w, ux = vy, uy = −wx. (3.25)

Then,P(0,0) =

(x, y, u, v, w) ∈ R2 × R3 : u = 0, v = w

,

andP(1,1) = J1P(0,0) ∩ P =

(x, y, 0, v, v, 0, 0, 0, 0, 0, 0) ∈ J1 pr 1

is the integral part of P . Thus, holonomic integral sections of P are of the form

φ(x, y) = (x, y, 0, c, c),

where c is any real number.

3.1.4 The Dual Jet Bundle

Denition 3.23. The dual jet bundle of π, denoted J1π†, is the reunion of the anemaps from J1

uπ to Λmπ(u)M , where u is an arbitrary point of E. Namely,

J1π† :=⋃u∈E

Aff(J1uπ,Λ

mπ(u)M). (3.26)

3.1. JET BUNDLES 31

The functions given byπ†1 : J1π† −→ Mω ∈ J1

uπ† 7−→ π(u)

(3.27)

andπ†1,0 : J1π† −→ Eω ∈ J1

uπ† 7−→ u

(3.28)

where J1uπ† = Aff(J1

uπ,Λmπ(u)M), are called the dual source projection and the dual target

projection respectively.

The duality nature of J1π† gives rise to a natural pairing between its elements andthose of J1π. The pairing will be denoted by the usual angular brackets, 〈 , 〉 : J1π† ⊗EJ1π → ΛmM .

Proposition 3.24. The dual jet bundle of π, J1π†, may be endowed with a structureof smooth manifold. A system of adapted coordinates (xi, uα) in E induces a system ofcoordinates (xi, uα, p, piα) in J1π† such that, for any j1

xφ ∈ J1π and any ω ∈ J1φ(x)π

†,xi(ω) = xi(x), uα(ω) = uα(φ(x)) and 〈ω, j1

xφ〉 = (p+ piαuαi ) dmx.

In the induced local coordinates (xi, uα, p, piα), the dual source and the dual targetprojections are written

π†1(xi, uα, p, piα) = (xi) and π†1,0(xi, uα, p, piα) = (xi, uα). (3.29)

From here, it is clear that π†1 and π†1,0 are certainly projections overM and E respectively.

Therefore, (J1π†, π†1,M) and (J1π†, π†1,0, E) are ber bundles. If we consider a change ofcoordinates (xi, uα) 7→ (yj, vβ) in E, it induces a change of coordinates (xi, uα, p, piα) 7→(yj, vβ, q, qjβ) in J1π†. In this case, the momenta transform by the following rule:

q = Jac(x(y))

(p+

∂uα

∂yjpiα∂yj

∂xi

)and qjβ = Jac(x(y))

(∂uα

∂vβpiα∂yj

∂xi

), (3.30)

where Jac(x(y)) is the Jacobian determinant of the transformation (yj) 7→ (xi). Note thatthe local volume form and its contraction transforms under the change of coordinates by

dmy = Jac(y(x)) dmx and dm−1yj = Jac(y(x))∂xi

∂yjdm−1xi. (3.31)

Proposition 3.25. The dual jet bundle of π, J1π†, together with the dual target projec-tion, π†1,0, is a vector bundle over E. Moreover, the induced coordinate systems (xi, uα, p, piα)are adapted to the vector bundle structure.

Denition 3.26. The reduced dual jet bundle of π, denoted J1π, is the quotient of J1π†

by constant ane transformations along the bers of π1,0. The quotient map will beµ : J1π† → J1π.

Proposition 3.27. We have that:

1. J1π may be endowed with a structure of smooth manifold;

2. (J1π†, µ, J1π) is a smooth vector bundle of rank 1;


3. Adapted coordinates (xi, uα) on E induce coordinates (xi, uα, piα) on J1π such thatµ(xi, uα, p, piα) = (xi, uα, piα), where (xi, uα, p, piα) are the induced coordinates onJ1π†.

The extended and the reduced dual jet bundles of π may also be realized by meansof basic and semi-basic forms. Recall that π-basic (resp. π-semi-basic) forms are formsover E annihilated by the contraction with at least one (resp. two) π-vertical vector.

Proposition 3.28. The extended dual jet bundle, J1π†, and the set of π-semi-basic m-forms over E, Λm

2 E, with canonical projection ΛkπE|Λk2E : Λk2E → E are isomorphic.

Proof. Given a semi-basic m-form ω ∈ Λm2 E, let u = ΛkπE(ω) ∈ E and consider the

function that sends any 1-jet j1xφ ∈ J1

uπ to the pullback of ω by φ at x. This does notdepend on the representative φ ∈ Γxπ of j1

xφ. Moreover, this function is ane withrespect to j1

xφ thus, this denes a morphism Υ from Λm2 E to J1π† as follows

Υ : Λm2 E −→ J1π†

ω 7−→ Υ(ω) : J1uπ −→ ΛmMj1xφ 7−→ φ∗xω

where u = π†1,0(ω). It is easy to check that Φ is a smooth isomorphism of vector bundles.

Semi-basic m-forms ω ∈ Λm2 E are locally written

ω = p dmx+ piα duα ∧ dm−1xi.

Thus, adapted coordinates (xi, uα) on E induce adapted coordinates on (xi, uα, p, piα) onΛm

2 E. The isomorphism dened in the previous proof takes then the local expression

Υ(xi, uα, p, piα) = (xi, uα, p, piα) : (xi, uα, uαi ) ∈ J1uπ 7→ p+ piαu

αi ∈ R.

Consider now the set of π-basic forms, Λm1 E. An arbitrary basic form ω is locally

writtenω = p dmx.

Notice that Λm1 E coincides with the pullback to E of ΛmM or with the set of constant

ane transformations on the bers of π1,0.

Corollary 3.29. The reduced dual jet bundle J1π is canonically isomorphic to the quo-tient of semi-basic m-forms Λm

2 E by the basic m-forms Λm1 E, that is J

1π ∼= Λm2 E/Λ

m1 E.

Proof. Let Ψ : Λm2 E → J1π† be the canonical isomorphism given in Proposition 3.28.

Since the set of constant ane transformations on the bers of π1,0 coincides with theset of basic m-forms over E, µ Ψ is constant along the bers of µ as well as Ψ µalong the bers of µ. Hence Ψ passes smoothly to the quotient to an isomorphismυ : Λm

2 E/Λm1 E → J1π.

While J1π† is naturally paired with J1π, remember that Λm2 E has a canonical mul-

tisymplectic structure (see Example 1.43). Consider the Liouville m-form Θ on Λm2 E,

which is locally given by the expression

Θ = p dmx+ piα duα ∧ dm−1xi, (3.32)

3.2. CLASSICAL FIELD THEORY 33

for adapted coordinates (xi, uα, p, piα) on Λm2 E. Then, the canonical multi-symplectic

(m+ 1)-form on Λm2 E is

Ω = − dΘ = − dp ∧ dmx− dpiα ∧ duα ∧ dm−1xi. (3.33)

Thanks to the identication between J1π† and Λm2 E (and their respective quotients),

any structure carried by one of them can be translated to the other. In particular, themulti-symplectic form. From now on, no distintion will be made between J1π† and Λm

2 E(or between J1π and Λm

2 E/Λm1 E). Although the dual notation will be used for sets,

coordinates, structures, etc.

3.2 Classical Field Theory

3.2.1 The Lagrangian Formalism

This section is devoted to the rst order Lagrangian formalism in jet manifolds. Themain ingredients are the following: the Lagrangian density, the Poincaré-Cartan form,the premultisymplectic structure dened from the multimomentum Liouville form andthe Legendre transformation. We shall use the same notations as in the previous section.

The variational approach

Denition 3.30. A Lagrangian density is a bered mapping L : J1π → ΛmM .

Since we assume that M is an oriented manifold, with volume form η, we can writeL = Lη, where L : J1π → R is the Lagrangian function. The manifold J1π plays the roleof the nite-dimensional conguration space of elds.

Denition 3.31. Given a Lagrangian density L : J1π −→ ΛmM , the associated integralaction is the map AL : Γπ ×K −→ R given by

AL(φ,R) =

∫R

(j1φ)∗L, (3.34)

where K is the collection of smooth compact regions of M .

Denition 3.32. Let φ be a section of π. A (vertical) variation of φ is a curve ε ∈ I 7→φε ∈ Γπ (for some interval I ⊂ R containing the 0) such that φε = ϕε φ (ϕε)

−1, whereϕε is the ow of a (vertical) π-projectable vector eld ξ on E.

Denition 3.33. We say that φ ∈ Γπ is a critical or stationary point of the Lagrangianaction AL if and only if

d

dε[AL(φε, Rε)]

∣∣∣ε=0

=d

dε

[∫Rε

(j1φε)∗L] ∣∣∣∣

ε=0

= 0, (3.35)

for any variation φε of φ whose associated vector eld vanishes outside of π−1(R).

Lemma 3.34. Let φε = ϕε φ (ϕε)−1 be a variation of a section φ ∈ Γπ. If ξ denotes

the innitesimal generator of ϕε, then

d

dε

[((j1φε) ϕε)∗xω

] ∣∣∣ε=0

= (j1φ)∗x(Lξ(1)ω), (3.36)

for any dierential form ω ∈ Ω(J1π).


Proof. From Proposition 3.16, we have that ξ(1) is the innitesimal generator of j1ϕε. Wethen obtain by a direct computation,

(j1φ)∗x(Lξ(1)ω) = (j1φ)∗x

(d

dε

[(j1ϕε)

∗ω] ∣∣∣ε=0

)=

d

dε

[(j1ϕε j1φ)∗xω

] ∣∣∣ε=0

=d

dε

[(j1φε j1ϕε)

∗xω] ∣∣∣ε=0

.

Theorem 3.35 (The Euler-Lagrange equations). Given a ber section φ ∈ Γπ, let usconsider an innitesimal variation φε of it such that the support R of the associatedvector eld ξ ∈ X(E) is contained in a coordinate chart (xi) of M . We then have that thevariation of the Lagrangian action AL at φ is given by

d

dεAL(φε, R)

∣∣∣ε=0

=

∫R

(j2φ)∗[(ξα − uαi ξi)

(∂L

∂uα− d

dxi∂L

∂uαi

)]dmx

−∫∂R

(j1φ)∗[ξiL+ (ξα − uαi ξi)

∂L

∂uαi

]dm−1xi,

(3.37)

where (xi, uα, uαi , uαij) are adapted coordinates on J2π. Moreover, φ is a critical point of

the Lagrangian action AL if and only if it satises the Euler-Lagrange equations

(j2φ)∗(∂L

∂uα− d

dxi∂L

∂uαi

)= 0 (3.38)

on the interior of M , plus the boundary conditions

(j1φ)∗L = (j1φ)∗∂L

∂uαi= 0, (3.39)

on the boundary ∂M of M .

Proof. Let us denote by ξ the vector eld associated to the variation φε. By Proposition3.34 and Cartan's formula L = d i+ i d, we have that

d

dεAL(φε, Rε)

∣∣∣ε=0

=

∫R

d

dε

[(j1φε ϕε)∗L

] ∣∣∣ε=0

=

∫R

(j1φ)∗(Lξ(1)L)

=

∫R

(j1φ)∗ d(iξ(1)L) +

∫R

(j1φ)∗iξ(1) dL

=

∫∂R

(j1φ)∗iξ(1)L+

∫R

(j1φ)∗(ξ(1)(L) dmx− dL ∧ iξ(1) dmx).

So as to develop the last three terms, we shall use the coordinate expression of ξ(1) givenin Proposition 3.15. Therefore, the rst boundary integral is∫

∂R

(j1φ)∗iξ(1)L =

∫∂R

(j1φ)∗(Lξi dm−1xi

).


For the second term, taking into account Equation (3.16) and using integration by parts,we obtain∫

R

(j1φ)∗[ξ(1)(L) dmx] =

=

∫R

(j1φ)∗[ξi∂L

∂xi+ ξα

∂L

∂uα+

(dξα

dxi− uαj

dξj

dxi

)∂L

∂uαi

]dmx

=

∫R

(j2φ)∗[ξi∂L

∂xi+ ξα

∂L

∂uα+

d

dxi(ξα − uαj ξj

) ∂L∂uαi

+ uαjiξj ∂L

∂uαi

]dmx

=

∫R

(j2φ)∗[ξi∂L

∂xi+ ξα

∂L

∂uα−(ξα − uαj ξj

) d

dxi∂L

∂uαi+ uαjiξ

j ∂L

∂uαi

]dmx

+

∫∂R

(j1φ)∗[(ξα − uαj ξj

) ∂L∂uαi

]dm−1xi.

And the third term is∫R

(j1φ)∗( dL ∧ iξ(1) dmx) =

=

∫R

(j1φ)∗[(

∂L

∂xidxi +

∂L

∂uαduα +

∂L

∂uαiduαi

)∧ ξj dm−1xj

]=

∫R

(j2φ)∗[ξi(∂L

∂xi+ uαi

∂L

∂uα+ uαji

∂L

∂uαj

)]dmx.

Adding the three developments that we have computed, some terms cancel out and,rearranging properly the remaining ones, we obtain the rst statement of the theorem.

If we now suppose that R is contained in the interior of M , as ξ is null outside of R,so it is ξ(1) outside of R and, by smoothness, on its boundary ∂R. Thus, if φ is a criticalpoint of AL, we then must have that

d

dεAL(φε, R)

∣∣∣ε=0

=

∫R


(∂L

∂uα− d

dxi∂L

∂uαi

)]dmx = 0,

for any vertical eld ξ whose compact support is contained in π−1(R). We thus infer thatφ shall satisfy the higher-order Euler-Lagrange equations (3.38) on the interior of M .

Finally, if R has common boundary with M and φ is a critical point of AL, from theabove results, we have that

d

dεAL(φε, R)

∣∣∣ε=0

= −∫∂R∩∂M

(j1φ)∗[ξiL+ (ξα − uαi ξi)

∂L

∂uαi

]dm−1xi = 0.

Since this is true for any vector eld ξ whose compact support is contained in π−1(R),then the boundary conditions (3.39) follows.

Remark 3.36. In the denition 3.33 of critical point of the Lagrangian action AL, we haveconsidered the widest range of variations, with the consequent decrement of the set ofpossible solutions. There, two dierent requirements on the variations could have beenmade, deriving in a broader set of solutions. First, we could have imposed verticality to


the variations, resulting in a substantial simplication of the proof, and we would stillhave obtained the Euler-Lagrange equations (3.38) but, this time, without the restriction(j1φ)∗L = 0 on ∂M . The same set of solution would have been obtained with verticalityonly along the boundary ∂M . Secondly, we could have imposed null variations along ∂M ,which would have implied no restrictions of the solutions of the Euler-Lagrange equationsalong ∂M , neither (j1φ)∗L = 0 nor (j1φ)∗∂L/∂uαi = 0.

If we have avoided these assumptions and followed this more general procedure is tostress out the strong relation with the geometric structure of jet bundles, in particularwith the so-called Poincaré-Cartan form, which will appear clear in the next section.

The geometric approach

Denition 3.37. The Poincaré-Cartan m-form associated with the Lagrangian densityL : J1π → ΛmM is dened by

ΘL := L+ 〈S, dL〉 , (3.40)

where S is the canonical vertical endomorphism of J1π and 〈S, dL〉 is the contractionbetween the factors in V π1 of S and those in T ∗J1π of dL. The Poincaré-Cartan (m+1)-form associated with L is dened by

ΩL := − dΘL. (3.41)

In local coordinates, if L = L dmx, we get:

ΘL =

(L− uαi

∂L

∂uαi

)dmx+

∂L

∂uαiduα ∧ dm−1xi (3.42)

= L+∂L

∂uαiθα ∧ dm−1xi,

ΩL = − d

(L− uαi

∂L

∂uαi

)∧ dmx− d

(∂L

∂uαi

)∧ duα ∧ dm−1xi (3.43)

= −θα ∧(∂L

∂uαdmx− d

(∂L

∂uαi

)∧ dm−1xi

).

If we conveniently denote piα := ∂L∂uαi

and p := L− piαuαi , then the local expression of thePoincaré-Cartan forms are now

ΘL = p dmx+ piα duα ∧ dm−1xi

ΩL = − dp ∧ dmx− dpiα ∧ duα ∧ dm−1xi

which are formally the expression of the Liouville forms of J1π† (compare with equations(3.32) and (3.33)). This is not a mere coincidence but, as we will see, an evidence of thestrong relation between the Lagrangian and the Hamiltonian formalisms.

Remark 3.38. Instead of using the canonical vertical endomorphism S in the denition3.37 of the Poincaré-Cartan m-form ΘL, we could have used the vertical endomorphismSη associated to a volume form η on M . If we dene

ΘL := L+ 〈Sη, dL〉 , (3.44)

where L = Lη and 〈Sη, dL〉 is the contraction between the factors in TJ1π of Sη andthose in T ∗J1π of dL, it turns out that this denition does not depend on the chosenvolume form η and coincides with the previous denition.


Remark 3.39. The Poincaré-Cartan m-form ΘL is also called DeDonder form by Binz,niatycki and Fisher [21], since this was the name used by Cartan (who attributed itsconstruction to DeDonder) to distinguish it from the Poincaré-Cartan form in Mechanics.In the quoted book by Binz et al. the reader can nd interesting historical remarksconcerning Field theories.

Proposition 3.40. The Poincaré-Cartan forms satisfy the following properties:

1. The Poincaré-Cartan m-form ΘL is π1,0-semi-basic, i.e. it is annihilated by anyπ1,0-vertical vector X ∈ V π1,0,

iXΘL = 0. (3.45)

2. The Poincaré-Cartan m-form ΘL is annihilated by any pair of π1-vertical vectorsX, Y ∈ V π1,

iXiY ΘL = 0. (3.46)

3. The Poincaré-Cartan (m + 1)-form ΩL is annihilated by any pair of π1,0-verticalvectors X, Y ∈ V π1,0,

iXiY iZΩL = 0. (3.47)

4. The Poincaré-Cartan (m + 1)-form ΩL is annihilated by any triple of π1-verticalvectors X, Y, Z ∈ V π1,

iXiY iZΩL = 0. (3.48)

5. Let ξ be a vector eld on E, we then have that

(j1φ)∗Lξ(1)L = (j1φ)∗Lξ(1)ΘL. (3.49)

Theorem 3.41. A section φ ∈ Γπ is a critical point of the Lagrangian action AL if andonly if its rst prolongation satises

(j1φ)∗(iξΩL) = 0, (3.50)

for any vector eld ξ ∈ J1π.

Lemma 3.42. Given a section σ ∈ Γπ1,0 and a vector eld ξ ∈ X(J1π) tangent to imσ,we have that

σ∗(iξΩL) = 0.

Proof. Along the image of σ, ξ shall have the form ξ = Tσ(v) for some vector eldv ∈ X(M). Then,

σ∗(iξΩL) = σ∗(iTσ(v)ΩL) = ivσ∗(ΩL) = 0,

since σ∗(ΩL) is an (m+ 1)-form on M which has dimension m.

Lemma 3.43. Given a section φ ∈ Γπ and a π1,0-vertical vector eld ξ ∈ X(J1π), wehave that

(j1φ)∗(iξΩL) = 0.


Proof. Let ξ = ξαi ∂/∂uαi be the local expression of the π1,0-vertical vector eld ξ. Then,

thanks to the local expression (3.43) of ΩL, we get

iξΩL = −ξαi∂2L

∂uαi ∂uβj

(θβ ∧ dm−1xj),

which is annihilated by j1φ since θβ is contact.

Proof of Theorem 3.41. Let φε be a vertical variation with compact support R ⊂M of asection φ ∈ Γπ and let ξ be its innitesimal generator. By Proposition 3.40 and Cartan'sformula, we have

d

dεAL(φε, R)

∣∣∣ε=0

=

∫R

(j1φ)∗(Lξ(1)L)

=

∫R

(j1φ)∗(Lξ(1)ΘL)

=−∫R

(j1φ)∗(iξ(1)ΩL) +

∫∂R

(j1φ)∗(iξ(1)ΘL).

Thus, using the Euler-Lagrange equations, Theorem 3.35, and the local expression(3.42) of the Poincaré-Cartan form ΘL, we deduce that∫

R

(j1φ)∗(iξ(1)ΩL) = −∫R


(∂L

∂uα− d

dxi∂L

∂uαi

)]dmx.

Therefore, φ satises the Euler-Lagrange equations (3.38) if and only if

(j1φ)∗(iξ(1)ΩL) = 0,

for any compactly supported π-projectable vector eld ξ ∈ X(E). Using partitions of theunity, we may generalize this to any π-projectable vector eld ξ ∈ X(E).

Finally, any vector eld ξ ∈ X(J1π) may be split into the sum of a vector eld onJ1π tangent to the image of j1φ, the rst lift to J1π of a vector eld on E and a π1,0-projectable vector eld on J1π. The assertion of the theorem follows from the previouslemmas.

Denition 3.44. The DeDonder equation is the following equation in terms of sectionsσ of π1 : J1π →M :

σ∗(iξΩL) = 0, ∀ξ ∈ X(J1π). (3.51)

Using Lemma 3.42, we have that a section σ ∈ Γπ1 still satises the DeDonder equationif we only consider π1-vertical vector elds ξ ∈ X(J1π). Taking this into account, an easycomputation using the local expression (3.43) of ΩL shows that the DeDonder equationis locally written

∂L

∂uα− ∂2L

∂xi∂uαi− ∂σβ

∂xi∂2L

∂uβ∂uαi−∂σβj∂xi

∂2L

∂uβj ∂uαi

+

(∂σβ

∂xi− uβi

)∂2L

∂uβi ∂uα

= 0,(3.52)(∂σβ

∂xj− uβj

)∂2L

∂uβj ∂uαi

= 0.(3.53)


Denition 3.45. A Lagrangian density L : J1π → ΛmM is regular whenever its Hessianwith respect to the velocities (

∂2L

∂uβj ∂uαi

)is non-degenerate.

Proposition 3.46. Let L : J1π → ΛmM be a regular Lagrangian density. A sectionσ ∈ Γπ1 satises the DeDonder equation if and only if σ is holonomic, i.e. σ = j1φ forsome φ ∈ Γπ, and φ satises the Euler-Lagrange equations.

Proposition 3.47. The Poincaré-Cartan (m+ 1)-form ΩL is multisymplectic, wheneverm > 1, and cosymplectic (together with the volume form η), whenever m = 1, if and onlyif the Lagrangian density L is regular.

Proof. Provided that m > 1, let (xi, uα, uαi ) be adapted coordinates on J1π. A straight-forward computation shows that

i ∂

∂xjΩL = (. . . )− ∂2L

∂uγk∂uαi

duγk ∧ duα ∧ dm−2xij

i ∂

∂uβΩL = (. . . ) +

∂2L

∂uβj ∂uαi

duαi ∧ dm−1xj

i ∂

∂uβj

ΩL = (. . . )− ∂2L

∂uβj ∂uαi

duα ∧ dm−1xi

where the indicated terms are the only ones with the correspondingm-form. Thus, assumethat v ∈ TJ1π is such that iξΩL = 0 and it is locally written in the given coordinates

ξ = ξj∂

∂xj+ ξβ

∂

∂uβ+ ξβj

∂

∂uβj.

If L is regular, then all coecients of ξ must be zero and ΩL is multisymplectic. Re-ciprocally, if ΩL is multisymplectic, then the Hessian of L has trivial kernel, i.e. L isregular.

For the case m = 1, consider coordinates (t, qα, vα) on J1π. In these coordinates, afterEquation (3.43), the Poincaré-Cartan (m+ 1)-form has the form

ΩL = − d

(∂L

∂vα

)∧ dqα + vα d

(∂L

∂vα

)∧ dt− ∂L

∂qαdqα ∧ dt.

A straightforward computation shows

ΩnL ∧ dt = det

(∂2L

∂vα∂vβ

)dq1 ∧ dv1 ∧ · · · ∧ dqn ∧ dvn ∧ dt,

which is a volume form if and only if L is regular.


Connections and multivector elds

Let Γ be a a connection in the bration π1 : J1π →M with horizontal projector h. So his locally expressed as follows:

h = dxi ⊗(∂

∂xi+ Aαi

∂

∂uα+ Aαji

∂

∂uαj

). (3.54)

Univocally associated to this connection, there is a class of locally decomposable multi-vector elds D(X) ⊂ Xm

d (J1π) locally expressed as follows (see Section 1.2):

X = f

m∧i=1

Xi = f

m∧i=1

(∂

∂xi+ Aαi

∂

∂uα+ Aαji

∂

∂uαj

). (3.55)

Proposition 3.48. Consider the dynamical equations

ihΩL = (m− 1)ΩL (3.56)

in terms of horizontal projectors h of connections Γ in the bration π1 : J1π →M , and

iXΩL = 0 (3.57)

in terms of locally decomposable m-multivector elds X ∈ Xmd (J1π). We have that both

equation are locally written

∂L

∂uα− ∂2L

∂xi∂uαi− Aβi

∂2L

∂uβ∂uαi− Aβji

∂2L

∂uβj ∂uαi

+(Aβi − u

βi

) ∂2L

∂uβi ∂uα

= 0, (3.58)

(Aβj − u

βj

) ∂2L

∂uβj ∂uαi

= 0, (3.59)

where (xi, uα, uαi ) are adapted coordinates on J1π and the A's are the coecients of h andX given in (3.54) and (3.55), respectively. It turns out that, if h and X are associated,then h satises (3.56) if and only if X satises (3.57).

Moreover, if Γ and/or X are integrable, then they satisfy the previous equations if andonly if its integral sections σ ∈ Γπ1 satisfy the DeDonder equation.

Proof. Using the local expression (3.43), we obtain on the one hand

ihΩL = (m− 1)ΩL +

[(Aβj − u

βj

) ∂2L

∂uβj ∂uαi

]duαi ∧ dmx

+

[∂L

∂uα− ∂2L


∂2L


∂2L

∂uβj ∂uαi

+(Aβj − u

βj

) ∂2L

∂uβj ∂uα

]duα∧ dmx.

On the other hand, a cumbersome computation1 yields

(−1)miXΩL =

[(∂piα∂xi− ∂p

∂uα

)+ Aβj

(∂pjα∂uβ−∂pjβ∂uα

)+ Aβji

∂piα

∂uβj

]︸︷︷︸

λα

duα

−

[∂p

∂uαi+ Aβj

∂pjβ∂uαi

]︸︷︷︸

λiα

duαi −(Aαkλα − Aαikλiα

]dxk,


where piα = ∂L∂uαi

and p = L− uαi piα, and where we have assumed that f = 1.

We deduce form here that, if h or X satisfy the corresponding equations (3.56) and(3.57), then their coecients must satisfy equations (3.58) and (3.59). The rst assertionof the theorem is now clear.

For the second statement, suppose that h and/or X are integrable and let σ ∈ Γπ1

be an integral section of any of them. Then, we have that ∂σα

∂xi= Aαi and

∂σαj∂xi

= Aαji,what yields to the local expressions (3.52) and (3.52) for σ of the DeDonder equation(3.51).

Notice that σ being an integral section of h or X does not mean that σ is holonomic,which is the case when L is regular as Proposition 3.46 assures.

Proposition 3.49. Let L : J1π → ΛmM be a regular Lagrangian density. Then thereexists a semi-holonomic connection Γ in π1 : J1π →M satisfying

ihΩL = (m− 1)ΩL, (3.60)

where h is the horizontal projector of Γ. Such a connection Γ will be called an Euler-Lagrange connection for L.

Proof. Given a locally nite open covering U1λλ∈Λ be of J1π with bered coordinates,

let αλλ∈Λ be a partition of the unity subordinate to U1λλ∈Λ. For each λ ∈ Λ, we dene

a horizontal projector hλ on U1λ as follows: Assuming that hλ must be described as in the

local expression (3.54), we take Aαj = uαj and we determine Aαij by means of the equation(3.58).

Denote by vλ = IdTJ1π−hλ the vertical projector and extend it by zero

vλ(u) =

αλ(u)vλ(u) if u ∈ supp(αλ)

0 if u /∈ supp(αλ)

for any u ∈ J1π. Now, we put

v(u) =∑λ∈Λ

vλ(u).

A direct computation shows that

im(vλ(u)) =

im(vλ(u)) if u ∈ supp(αλ)

0 if u /∈ supp(αλ)

1 Here, we have used the formulae

(−1)miX(α ∧ dmx) = α− α(Xj) dxj ,

(−1)miX(β ∧ αi ∧ dm−1xi) = β(Xj)αj(Xi) dx

i − β(Xj)αi(Xi) dx

j − β(Xj)αj + αj(Xj)β,

where α, αi and β are 1-forms and where X is the m-vector X =∧m

i=1Xi.


from which one deduces that im(v(u)) ⊆∑

λ∈Λ im(vλ(u)) ⊆ Vu π1. Furthermore, we have

v(u)|Vu π1 =∑λ∈Λ

vλ(u)|Vu π1

=∑λ∈Λ

αλ(u)vλ(u)|Vu π1

=∑λ∈Λ

αλ(u) IdTJ1π |Vu π1

= IdTJ1π |Vu π1 .

So we deduce that v is a globally well dened vertical projector over J1π, thus it inducesa semi-holonomic connection Γ in π1 : J1π →M (by construction) satisfying (3.60).

Corollary 3.50. Let L : J1π → ΛmM be a regular Lagrangian density. Then there existsa semi-holonomic multivector eld X ∈ Xm

d (J1π) satisfying

iXΩL = 0. (3.61)

Such a connection multivector eld X will be called an Euler-Lagrange multivector forL.

Remark 3.51. In order to discuss the uniqueness, suppose that Γ1 and Γ2 are two solutionsof (3.60). If we denote by T the tensor eld T = h1−h2, dierence of the two horizontalprojectors then, using that Γ1 and Γ2 are semi-holonomic, we deduce that T is locally

given by T = Tαij∂

∂uαi⊗ dxj. Moreover, iT ΩL = 0 implies that

Tαij∂2L

∂uαi ∂uβj

= 0 ,

for all β ∈ 1, . . . , n. Since we have n equations and nm2 unknowns, the solutions ateach point form a vector space of dimension nm2 − n, taking into account the regularityassumption on L. Therefore, the solutions of 3.60 are given by h + T , where h is thehorizontal projector of a particular solution, T is a tensor eld of type (1,1) on J1πsuch that it takes values in the vertical bundle V π1,0, it vanishes when it is applied toπ1-vertical vector elds and iT ΩL = 0. In fact, the space T of all tensor elds T isa C∞(J1π)-module with local dimension n(m2 − 1). If dimM = 1, then there exists aunique solution ΓL of (3.60).

3.2.2 The Hamiltonian Formalism

See [31, 68, 69, 73, 75, 77, 91, 92, 115, 133, 134, 136, 137].

Denition 3.52. A Hamiltonian section is a section h : J1π → J1π† of µ : J1π† → J1π.

Denition 3.53. A Hamiltonian density is a smooth function H : J1π† → ΛmM suchthat iξ dH = iξΩ for any µ-vertical vector eld ξ ∈ V µ.


In coordinates (xi, uα, p, piα), a Hamiltonian section h ∈ Γµ is locally described by

h(xi, uα, piα) = (xi, uα, p = −H(xi, uα, piα), piα), (3.62)

where the smooth function H, which is locally dened, is called the Hamiltonian function.Given a Hamiltonian density H, let ξ ∈ V µ be a µ-vertical vector eld. In local

coordinates (xi, uα, p, piα), ξ shall has the form ξ = ξ0∂/∂p, for some locally denedfunction ξ0 on J1π†. In order to satisfy the denition, we must have

iξ dH = iξ( dH ∧ dmx) = ξ0∂H

∂pdmx = −ξ0 dmx = iξΩ,

where H = Hη. Since this shall be true for any ξ ∈ V µ, we have that Hamiltoniandensity H is in turn locally described by

H(xi, uα, p, piα) = p+H(xi, uα, piα). (3.63)

The smooth function H coincides with the previous Hamilton function in the followingsense.

Proposition 3.54. The space of Hamiltonian sections and the family of Hamiltoniandensities are in bijective correspondence. In fact, a Hamiltonian section h and a Hamil-tonian density H are univocally related by the condition imh = H−1(0). In this case, wesay that they are associated.

By means of this relation, we may relate also section of π1 and π†1.

Corollary 3.55. Let h : J1π → J1π† be a Hamiltonian section associated with a Hamil-tonian density H ∈ Ωm(π†1). A section σ ∈ Γπ1, denes a section σ = h σ ∈ Γπ†1such that σ∗H = 0. Reciprocally, a section σ ∈ Γπ†1 with σ∗H = 0 denes a sectionσ = µ σ ∈ Γπ1 such that σ = h σ. In both cases, we say that σ and σ are associated.

Denition 3.56. Let h ∈ Γµ be a Hamiltonian section. The Cartan m-form associatedto h is dened by

Θh := h∗Θ. (3.64)

The Cartan (m+ 1)-form associated to h is dened by

Ωh := − dΘh = h∗Ω. (3.65)

It is worth to recall that the Liouville form and the canonical one are locally given by

Θ = p dmx+ piα duα ∧ dm−1xi, (3.66)

Ω = − dp ∧ dmx− dpiα ∧ duα ∧ dm−1xi; (3.67)

in adapted local coordinates (xi, uα, p, piα) on J1π†. Thus, in the corresponding inducedcoordinates (xi, uα, piα) on J1π, we have that the Cartan forms are given by

Θh = −H dmx+ piα duα ∧ dm−1xi, (3.68)

Ωh = dH ∧ dmx− dpiα ∧ duα ∧ dm−1xi. (3.69)


Proposition 3.57. Let H : J1π† → ΛmM be a Hamiltonian density associated with aHamiltonian section h ∈ Γµ. We have that

Θ = µ∗Θh +H and Ω = µ∗Ωh − dH. (3.70)

Proof. The second equation follows from the rst, which is immediate from the previouslocal expressions.

Denition 3.58. Let H : J1π† → ΛmM be a Hamiltonian density. The Cartan m-formassociated to H is dened by

ΘH := Θ−H. (3.71)

The Cartan (m+ 1)-form associated to H is dened by

ΩH := − dΘH = Ω + dH. (3.72)

The Hamilton equations

Denition 3.59. Given a Hamiltonian section h ∈ Γ, the associated (reduced) Hamilto-nian action is the map Ah : Γπ1 ×K → R given by

Ah(σ,R) =

∫R

σ∗Θh, (3.73)


Let h : J1π → J1π† be a Hamiltonian section associated with a Hamiltonian densityH ∈ Ωm(π†1). Given a section σ : M → J1π of π†1 : J1π† → M , the compositionσ = hσ : M → J1π† denes a section of π†1 : J1π† →M . Note that, in general, a sectionσ ∈ Γπ†1 does not dene a section σ ∈ Γπ1 such that σ = h σ, which is only true whenσ∗H = 0 (from Proposition 3.54). Besides, for this particular section σ ∈ Γπ1, we havethat

σ∗Θh = σ∗(µ∗Θh) = σ∗ΘH.

Thus, the extremals of Ah coincide through h with the extremals restricted to σ∗H = 0of the following integral action.

Denition 3.60. Given a Hamiltonian density H : J1π† → ΛmM , the associated (ex-tended) Hamiltonian action is the map AH : Γπ†1 ×K → R given by

AH(σ,R) =

∫R

σ∗ΘH, (3.74)


Theorem 3.61 (Hamilton's equations). Let H : J1π† → ΛmM be a Hamiltonian densityassociated with a Hamiltonian section h ∈ Γµ. Critical points of each integral action arecharacterized by the Hamilton equations plus boundary conditions, which are

σ∗(iξΩh) = 0 and σ∗(iξΘh) =∂M 0, ∀ξ ∈ X(J1π), (3.75)

for a critical point σ ∈ Γπ1 of Ah and

σ∗(iξΩH) = 0 and σ∗(iξΘH) =∂M 0, ∀ξ ∈ X(J1π†), (3.76)


for a critical point σ ∈ Γπ†1 of AH. Moreover, in both cases, the Hamilton equations havethe common local expression

∂H

∂uα= −∂σ

iα

∂xiand

∂H

∂piα=∂σα

∂xi, (3.77)

and the boundary conditions are

σ∗H =∂M 0 and σiα =∂M 0, (3.78)

where (xi, uα, p, piα) and (xi, uα, piα) are adapted coordinates on J1π† and J1π respectively,σ = (xi, σα, σiα) and σ = (xi, σα, σ0, σ

iα).

Proof. We begin by determining the variation of the reduced Hamiltonian action Ah.Given a section σ of π1 : J1π → M , let R be a compact region of M and σε = ϕε σ (ϕε)

−1 a variation of σ where the innitesimal generator ξ ∈ X(J1π) of ϕε has itssupport contained in (π1)−1(R). Applying a result similar to Lemma 3.34 and Cartan'sformula, we obtain

d

dε[Ah(σε, Rε)]

∣∣∣ε=0

=d

dε

[∫R

(ϕε σ)∗Θh

] ∣∣∣∣ε=0

=

∫R

σ∗(LξΘh)

= −∫R

σ∗(iξΩh) +

∫∂R

σ∗(iξΘh).

Thus, σ is a critical point of Ah if an only if∫R

σ∗(iξΩh)−∫∂R

σ∗(iξΘh) = 0

for any compact region R ⊆ M and any vector eld ξ ∈ X(J1π) whose support iscontained in (π1)−1(R).

Now, assume that σ is a critical point of Ah. If R is a compact region contained in theinterior of M , then any vector eld ξ ∈ X(J1π) whose support is contained in (π1)−1(R)must be null along the bers over the boundary of R. Indeed, for such R and σ, we have∫

R

σ∗(iξΩh) = 0.

Varying R and ξ, and using partitions of the unity, we deduce that

σ∗(iξΩh) = 0

for every vector eld ξ ∈ X(J1π).In a similar way, we deduce the boundary condition

σ∗(iξΘh) =∂M 0

for every vector eld ξ ∈ X(J1π).


Finally, let us compute the local expression of σ∗(iξΩh). Given adapted coordinates(xi, uα, piα) on J1π, if we denote σ = (xi, σα, σiα), we then have

σ∗(iξΩh) = σ∗[(

ξα∂H

∂uα+ ξiα

∂H

∂piα

)dmx−

(∂H

∂uαduα +

∂H

∂piαdpiα

)∧ ξj dm−1xj

− ξiα duα ∧ dm−1xi + ξα dpiα ∧ dm−1xi − dpiα ∧ duα ∧ ξj dm−2xij

]=

[− ξj

(∂σα

∂xj∂H

∂uα+∂σiα∂xj

∂H

∂piα+∂σiα∂xi

∂σα

∂xj− ∂σjα∂xi

∂σiα∂xj

)+ ξα

(∂H

∂uα+∂σiα∂xi

)+ ξiα

(∂H

∂piα− ∂σα

∂xi

)]dmx,

where we have used the relation

dxk ∧ dm−2xij = δkj dm−1xi − δki dm−1xj.

Provided σ is a critical point of Ah, since σ∗(iξΩh) must be null for any ξ ∈ X(J1π), wetherefore shall have that

∂H

∂uα= −∂σ

iα

∂xiand

∂H

∂piα=∂σα

∂xi,

which are precisely the local expression of the Hamilton equations.For the boundary condition σ∗(iξΘh) = 0 over ∂M , we proceed in the same way, and

we get that locally

σ∗(iξΘh) = σ∗(−Hξi − piαuαj ξj − pjαuαj ξi − piαξα) dm−1xi = 0,

which implies thatσ∗H = σiα = 0,

along ∂M .The proof of the theorem for the case of the extended Hamiltonian action AH is

completely analogous.

Remark 3.62. Even though the proof seems to be valid only for the multidimensionalcase (m > 1), because of the fact that the (m− 2)-form dm−2xij appears explicitly in thedevelopment of σ∗(iξΩh), it remains valid when m = 1. In fact, in this case, the termswith dm−2xij would disappear and dm−1xi = 1.

Remark 3.63. It follows from the derivation of the local expression of the Hamilton'sequations and the boundary conditions that the considerations made in Remark 3.36 arestill valid here. If we had restricted the variations to π1-vertical or π

†1-vertical ones over

the whole of M or only over ∂M , then only the boundary condition σ∗H = 0 along ∂Mwould have remained. Moreover, if we had considered null variations at the border ∂M ,then any boundary condition would had remained and we would be free to x them.

The main dierence between the reduced and the extended formalism is that, in theextended one, there are a wider number of critical sections since there are no restrictionson the component σ0 = p σ of a critical section σ. Nonetheless, critical sections of thereduced Hamiltonian action Ah are always critical sections of the extended Hamiltonianaction AH.


Corollary 3.64. A section σ ∈ Γπ1 is a critical point of the reduced Hamiltonian actionAh if and only if the associated section σ = h σ ∈ Γπ†1 is a critical point of the extendedHamiltonian action AH (and vice versa).

Proof. The proof is trivial using the coordinate expression (3.77) of the Hamilton equa-tions (3.75) and (3.76), or taking into account the relation (3.70) between the Cartanforms and that ΘH and ΩH are both µ-basic.

Let Γ be a a connection in the bration π†1 : J1π† → M with horizontal projector h.So h is locally expressed as follows:

h = dxj ⊗(

∂

∂xj+ Aαj

∂

∂uα+B i

αj

∂

∂piα+ Cj

∂

∂p

). (3.79)

Univocally associated to this connection, there is a class of locally decomposable multi-vector elds D(X) ⊂ Xm

d (J1π†) locally expressed as follows (see Section 1.2):

X = f

m∧j=1

Xj = f

m∧j=1

(∂

∂xj+ Aαj

∂

∂uα+B i

αj

∂

∂piα+ Cj

∂

∂p

). (3.80)

Let H : J1π† → ΛmM be a Hamiltonian density and h : J1π → J1π† the associatedHamiltonian section. A connection Γ in π1 : J1π → M (and any associated multi-vector eld X ∈ Xm

d (J1π)) induces a connection Γ in π†1 : J1π† → M (and associatedmultivector elds X ∈ Xm

d (J1π†)) along im(h) such that

Cj = −∂H∂xj− Aαj

∂H

∂uα−B i

αj

∂H

∂piα.

Proposition 3.65. Let H : J1π† → ΛmM be a Hamiltonian density and h : J1π → J1π†

the associated Hamiltonian section.

1. The (extended) dynamical equations

ihΩH = (m− 1)ΩH and iXΩH = 0 (3.81)

in terms of the horizontal projectors h of connections Γ in π†1 : J1π† → M andlocally decomposable multivector elds X ∈ Xm

d (J1π†) are equivalent whenever hand X are associated. Moreover, the integral sections σ of solutions h or X of theextended dynamical equations (3.81) are solutions of the extended Hamilton equation(3.76).

2. The (reduced) dynamical equations

ihΩh = (m− 1)Ωh and iXΩh = 0 (3.82)

in terms of the horizontal projectors h of connections Γ in π1 : J1π → M andlocally decomposable multivector elds X ∈ Xm

d (J1π) are equivalent whenever hand X are associated. Moreover, the integral sections σ of solutions h or X of thereduced dynamical equations (3.82) are solutions of the reduced Hamilton equation(3.75).


Proof. We prove only the extended case, since the reduced one is completely analogous.Under the given assumptions, we have on the one hand that

ihΩH = (m− 1)ΩH + ( dH +B iαi du

α − Aαi dpiα) ∧ dmx.

On the other hand, we have that

(−1)miXΩH =

(∂H

∂uα+B i

αi

)duα +

(∂H

∂piα− Aαi

)dpiα

+

(Aαi B

iαj − AαjB i

αi − Aαj∂H

∂uα+B i

αj

∂H

∂piα

)dxj.

Therefore, the dynamical equations (3.81) are written in terms of the coecients of hand X

∂H

∂uα= −B i

αi and∂H

∂piα= Aαi .

We deduce from here that the dynamical equations (3.81) are equivalent whenever h andX are associated.

If σ is an integral section of h or X, then B iαj = ∂σiα/∂x

j and Aαi = ∂σα/∂xi, and werecover the local Hamilton equations (3.77).

3.2.3 The Legendre transformation

Denition 3.66. Given a Lagrangian density L : J1π → ΛmM , the extended Legendretransformation is the map LegL : J1π → J1π† dened as follows: let j1

xφ ∈ J1π, for anym tangent vectors ξ1, . . . , ξm ∈ Tφ(x)E, then LegL(j1

xφ) gives

LegL(j1xφ)(ξ1, . . . , ξm) := (ΘL)j1xφ(ξ1, . . . , ξm), (3.83)

where ξi is any tangent vector to J1π at j1xφ that projects to ξi.

The reduced Legendre transformation is the map legL : J1π → J1π dened by legL :=µ LegL.

Recall that the Poincaré-Cartan form ΘL is π1,0-basic and π1-semi-basic (see Proposi-tion 3.40). We thus have that, in one hand, the Legendre transformation does not dependon the chosen vectors ξ1, . . . , ξm and, in the other hand, the image of LegL are π-semi-basic m-forms over E. Henceforth, the Legendre transformation LegL is well dened andgives values in J1π†. Furthermore, from the denition, both Legendre transformationsare clearly morphisms of ber bundles over the identity of E, which is also clear fromtheir local expressions

LegL(xi, uα, uαi ) =

(xi, uα, p = L− uαi

∂L

∂uαi, piα =

∂L

∂uαi

), (3.84)

legL(xi, uα, uαi ) =

(xi, uα, piα =

∂L

∂uαi

). (3.85)

Proposition 3.67. Let L : J1π → ΛmM be a Lagrangian density. The following state-ment are equivalent:

1. L is regular;


2. LegL : J1π → J1π† is an immersion;

3. legL : J1π → J1π is a local dieomorphism.

Denition 3.68. A Lagrangian density L : J1π → ΛmM is hiper-regular whenever legLis a global dieomorphism.

In such a case, we have that J1π, im(LegL) and J1π are dieomorphic. Moreover, h :=LegL leg−1

L is a Hamiltonian section and im(LegL) is the 0-level set of the Hamiltoniandensity associated to h.

Proposition 3.69. Let L : J1π → ΛmM be any Lagrangian density. Then, we have

Leg∗LΘ = ΘL and Leg∗LΩ = ΩL. (3.86)

Furthermore, if L is hiper-regular, we may dene the Hamiltonian section h := LegL leg−1L

and consider the Hamiltonian density H associated to h. We then have

Leg∗LΘH = ΘL and Leg∗LΩH = ΩL, (3.87)

leg∗LΘh = ΘL and leg∗LΩh = ΩL. (3.88)

Proof. The rst equation derives easily from the local expressions (3.42) of ΘL, (3.66) ofΘ and (3.84) of LegL. The others follows directly.

Theorem 3.70 (The equivalence theorem). Given a hiper-regular Lagrangian densityL : J1π → ΛmM , let h := LegL leg−1

L be the induced Hamiltonian section and H theHamiltonian density associated to h. If a section σ1 ∈ Γπ1 satises the DeDonder equation(3.51),

σ∗1(iξΩL) = 0, ∀ξ ∈ X(J1π),

then the sections σ2 = legL σ1 ∈ Γπ1 and σ2 = LegL σ1 ∈ Γπ†1 satises the correspond-ing Hamilton equations (3.75) and (3.76),

σ∗2(iξΩh) = 0, ∀ξ ∈ X(J1π),

and

σ∗2(iξΩH) = 0, ∀ξ ∈ X(J1π†).

Reciprocally, if σ2 ∈ Γπ1 (resp. σ2 ∈ Γπ†1 with σ∗2H = 0) satisfy the correspondingHamilton equation, then σ1 = leg−1

L σ2 ∈ Γπ1 (resp. σ1 = leg−1L µ σ2 ∈ Γπ1) satises

the DeDonder equation.

Remark 3.71. Observe that the Lagrangian boundary conditions (3.39) are transformedby the Legendre map to the Hamiltonian boundary conditions (3.78). Therefore, thevariations considered within the theory must be correspond properly in the Lagrangianand the Hamiltonian side as stated in Remark 3.36 and Remark 3.63.


3.2.4 The Skinner and Rusk formalism

What follows may be found in here [70, 50].

Denition 3.72. The bered product

W := J1π ×E J1π† (resp. W := J1π ×E J1π) (3.89)

is called the mixed space of velocities and extended ( resp. reduced) momenta. The canon-ical projections are denoted pr 1 : W → J1π and pr 2 : W → J1π† (resp. with abuse ofnotation pr 1 : W → J1π and pr 2 : W → J1π). The projections as a ber bundleover E and M are πW,E = π1,0 pr 1 and πW,E = π1 pr 1 (resp. πW ,E = π1,0 pr 1 andπW ,E = π1 pr 1). We still denote the canonical projection by µ : W → W .

We deduce from Propositions 3.3 and 3.24 that adapted coordinates (xi, uα) in Einduce adapted coordinates (xi, uα, uαi , p, p

iα) on W , where (uαi ) and (p, piα) are bered

coordinates on J1π → E and J1π† → E, respectively. Accordingly, we have adaptedcoordinates (xi, uα, uαi , p

iα) on W .

Wpr1

zzzzzzzz

πW,E

pr2

""EEEEEEEEµ //W

pr2

##HHHHHHHHH

J1ππ1,0

!!DDDDDDDD J1π†

||yyyyyyyy

µ // J1π

uukkkkkkkkkkkkkkkkk

E

π

M

The Liouville form Θ and the canonical multisymplectic form Ω of J1π† are pulledback to W by pr 2, which we continue denoting by the same letters. It should be noticedthat (W,Ω) is no longer multisymplectic, but pre-multisymplectic. We also have to ourdisposal the natural pairing natural pairing 〈 , 〉 : J1π† ×E J1π. Therefore, we have thebered map

W = J1π ×E J1π†Φ //

πW,E

ΛmM

E

π //M

where Φ := 〈pr 2, pr 1〉. If we realize J1φ† as the space of semi-basic m-forms over E (see3.28), then Φ takes the form

Φ(w) = φ∗x(ω), (3.90)

where w = (j1xφ, ω) ∈ W and φ†1,0(ω) = φ(x). In the local coordinates (xi, uα, uαi , p, p

iα) of

W , the internal pairing Φ is given by

Φ(xi, uα, uαi , p, piα) = (p+ piαu

αi ) dmx. (3.91)

Observe that we have used the map in the proof of the identication between J1π† andΛm

2 E.


Together with the pairing Φ and the pre-multisymplectic (m + 1)-form Ω, we intro-duce a Lagrangian density to dene a Hamiltonian density on W and, let us say, thecorresponding Cartan forms.

Denition 3.73. Assume that L : J1π −→ ΛmM is a Lagrangian density.

1. The Hamiltonian density (or the generated energy density following [151]) associatedto L on W is the map H : W → ΛmM dened by

H = Φ− L pr 1 . (3.92)

2. The Hamiltonian section associated to L on W is the unique section h : W → Wof µ : W → W whose image coincides with the 0-level set of H, i.e. such thatimh = H−1(0).

3. The Hamiltonian submanifold of W , let say W0, is identied with the 0-level set ofthe associated Hamiltonian density H or the image of the associated Hamiltoniansection h, that is,

W0 := w ∈ W : H(w) = 0 = im(h). (3.93)

In bered coordinates (uαi , p, piα) of W and (uαi , p

iα) of W , the Hamiltonian density,

section and submanifold are respectively given by

H(xi, uα, uαi , p, piα) = (p+ piαu

αi − L) dmx; (3.94)

h(xi, uα, uαi , piα) = (xi, uα, uαi , p = L− piαuαi , piα); (3.95)

W0 =

(xi, uα, uαi , p, piα) ∈ W : p = L− piαuαi

. (3.96)

From here, we may observe that H corresponds precisely to a Hamiltonian density inthe sense of Denition 3.53: For any vertical µ-vector eld ξ, we do have iξΩ = iξ dH.Even if it is obvious, it is worth to note that W and W0 are dieomorphic, being h thedieomorphism between them.

Denition 3.74. Given a Lagrangian density L : J1π −→ ΛmM . LetH be the associatedHamiltonian density and h ∈ Γµ the associated Hamiltonian section.

1. The Cartan m-form and (m+ 1)-form associated to H are

ΘH := Θ−H and ΩH := − dΘH = Ω− dH. (3.97)

2. The Cartan m-form and (m+ 1)-form associated to h are

Θh := h∗Θ and Ωh := − dΘh = h∗Ω. (3.98)

Following Proposition 3.57, one could check that

ΘH = µ∗Θh and ΩH = µ∗Ωh. (3.99)

In bered coordinates (uαi , p, piα) of W and (uαi , p

iα) of W , the Cartan forms are given by

ΘH =(L− piαuαi ) dmx+ piα duα ∧ dm−1xi (3.100)

ΩH =

(∂L

∂uαduα +

∂L

∂uαiduαi − uαi dpiα + piα duαi

)∧ dmx− dpiα ∧ duα ∧ dm−1xi, (3.101)


where L = L dmx. Hence Θh and Ωh have formally the same developments.As we have already stated, while (J1π†,Ω) was multisymplectic, (W,Ω) is only pre-

multisymplectic and so are (W,ΩH), (W0,ΩH|TW0) and (W ,Ωh).We are now in position to introduce the equation that establishes the eld dynamics

within the Skinner-Rusk formalism. As in the previous sections 3.2.1 and 3.2.2, wecould do it by means of horizontal projectors of a given connection or using the associatedmultivector eld. In this case, we will restrict to the method of horizontal projectors, butthe reader may check that the same equations will follow considering multivector elds.

Denition 3.75. The dynamical equation is the following equation in terms of horizontalprojectors h of the corresponding connections Γ in πW,M : W →M :

ihΩH = (m− 1)ΩH. (3.102)

As we are going to see, the previous equation is only solvable in a subset W ′1 of

W . If we require that W ′1 be a smooth submanifold of W and that the solutions be

horizontal projectors of connections along W ′1, we will end up with further restrictions on

the projectors and, whenever L is not regular, with further constraints on the manifoldalong which the connections are dened. This chain of consequences is known as theGotay-Nester-Hinds algorithm, although it was originally dened for classical mechanics.The submanifold W ′

1 is called the rst constraint manifold and it is obtained at the rststep of the algorithm. The nal constraint manifold W ′

f along which the solutions lie isobtained as a x point and nal step of the algorithm.

Theorem 3.76. The solutions of the dynamical equation (3.102) restricted to W0 are,in the best case, horizontal projectors of connections along a submanifold Wf of W0. Inparticular, if h is such a solution, which is assumed to be written in the form

h = dxj ⊗(

∂

∂xj+ Aαj

∂

∂uα+ Aαij

∂

∂uαi+B i

αj

∂

∂piα+ Cj

∂

∂p

), (3.103)

then it must satisfy the equation of holonomy

Aαi = uαi , (3.104)

the equations of dynamics

B jαj =

∂L

∂uα, (3.105)

piα = =∂L

∂uαi, (3.106)

plus the equations of tangency

B iαj =

∂2L

∂xj∂uαi+ uβ

∂2L

∂uβ∂uαi+ Aβkj

∂2L

∂uβk∂uαi

, (3.107)

Cj =∂L

∂xj+ uαj

∂L

∂uα−B i

αjuαi . (3.108)

The submanifold Wf is contained in the submanifold of W0 dened by

W1 :=

(xi, uα, uαi , p, p

iα) ∈ W : p = L− piαuαi , piα =

∂L

∂uαi

, (3.109)

and coincides with it when L is regular.


Proof. We begin by dening W1 as the subset of W where point-wise solutions of thedynamical equation (3.102) exist, that is

W ′1 := w ∈ W / ∃hw : TwW −→ TwW linear such that h2

w = hw,

kerhw = Vw πW,M , ihwΩH(w) = (m− 1)ΩH(w).

For a given point w ∈ W , we x a chart around it with coordinates (xi, uα, uαi , p, piα) and

consider an arbitrary horizontal projector hw in TWW . Then, hw must certainly have theform (3.103). We therefore compute

ihwΩH − (m− 1)ΩH =[(B iαi −

∂L

∂uα

)duα +

(piα −

∂L

∂uαi

)duαi + (uαi − Aαi) dpiα

]dmx,

equating to zero, we deduce that, in order to be a solution of the dynamical equation(3.102), hw must be dened over a point w that satises Equation (3.106) and its coe-cients the equations (3.104) and (3.105).

By a reasoning in terms of partitions of the unity similar to the one given in the proofof Proposition 3.49, we obtain a horizontal projector h : TW ′1W → TW ′1W dened overW ′

1 which satises the dynamical equation (3.102). We now restrict h to be dened overW1 := W0 ∩W ′

1, hence obtaining a horizontal projector h : TW1W → TW1W dened overW1 which satises the dynamical equation (3.102). But we still have to ensure that h isa horizontal projector along W1, that is h takes values in TW1: Therefore, we impose thetangency condition hw(TwW ) ⊂ TwW1, ∀w ∈ W1. This latter condition is equivalent tohaving

h

(∂

∂xj

)(piα −

∂L

∂uαi

)= 0 and h

(∂

∂xj

)(piα −

∂L

∂uαi

)= 0,

which in turn is equivalent (using the previous relations) to equations (3.107) and (3.108).By combining the rst equation of dynamics (3.105) with the rst equation of tangency

(3.107), we get∂L

∂uα− ∂2L

∂xi∂uαi− uβi

∂2L


∂2L

∂uβj ∂uαi

= 0.

If L is regular, nothing else can be stated than that Wf coincides with W1 and thath denes a connection along it. Otherwise, depending on the non-regularity of L, thelast equation could derive restrictions in W0 to obtain the rst constraint manifold (seeremark 3.77 below). Nevertheless, it is contained in the submanifold W1 given above.

Remark 3.77. It shall be said that Theorem 3.76 remains true when the dynamical equa-tion (3.102) is considered on the whole of W (instead of restricted to W0), but then W1

should be changed by W ′1, so the tangency condition (3.108) is no longer available.

We may note in the Theorem's proof that, while the coecients Aαi and Cj of h arecompletely determined (equations (3.104) and (3.108)), the coecients B i

αj are overdeter-mined (equations (3.105) and (3.107)), what gives an extra restriction on the coecientsAαij for each α:

∂L

∂uα− ∂2L


∂2L


∂2L

∂uβj ∂uαi

= 0. (3.110)


Although, the latter coecients cannot be completely determined in general.When the base manifold M has dimension m = 1, further restrictions could be ob-

tained on the manifold along which h is dened, depending on if the Lagrangian densityis regular or not. In this case, W1 would include these restrictions and they will givefurther tangency conditions to determine the coecients of h. Assume that M = 1 andlet (t, qα, vα, p, pα) denote adapted coordinates on W . By Theorem 3.76, a solution h ofthe dynamical equation (3.102) would satisfy in particular the equations

Bα =∂L

∂qαand Bα =

∂2L

∂t∂vα+ vβ

∂2L

∂qβ∂vα+ Aβ

∂2L

∂vβ∂vα,

where

h = dt⊗(∂

∂t+ vα

∂

∂qα+ Aα

∂

∂vα+Bα

∂

∂piα+ C

∂

∂p

).

Therefore, if L is not regular and we consider an element (V α) in the kernel of ∂2L∂vβ∂vα

,then (

∂L

∂qα− ∂2L

∂t∂vα− vβ ∂2L

∂qβ∂vα

)V α = 0,

which is a new restriction that determines the submanifold where h is dened.Analogously, if the base manifold is multidimensional (m > 1) then, a possible way

to obtain constraints derived from Equation (3.110) is to nd an non-trivial element V α

such that∂2L

∂uβj ∂uαi

V α = 0, ∀i, j, β,

but this is no an easy task. The new constraint would then be(∂L

∂uα− ∂2L


∂2L

∂uβ∂uαi

)V α = 0.

Unfortunately, this method cannot be used in the general case (m > 1 and n > 1).However, in both cases (m = 1 or m > 1), a remarkable fact is that the semi-

holonomy of h yields immediately (Equation (3.104)) whether the Lagrangian densityis regular or not, which diers from the Lagrangian formalism (see Proposition 3.49 orCorollary 3.50). Taking this into account, there is a clear analogy between Equation(3.110) and the equations derived in the proof of Proposition 3.48.

Example 3.78. Consider the ber bundle pr 1 : R4 → R2 with global adapted coordinates(x, y, u, v) and base volume form dx ∧ dy. We consider the Lagrangian function L :J1 pr 1 → R

L = uv + (ux + vx)(uy + vy).

In this case, Equation (3.110) reads

v − Auyx − Avyx − Auxy − Avxy = 0,

u− Auyx − Avyx − Auxy − Avxy = 0.

From where we deduce that u = v, hence the rst constraint submanifold is

W1 = w ∈ W : p0 = uv − pxqy, px = uy + vy = qx, py = ux + vx = qy, u = v .


Requiring that h be dened along, we obtain the second and nal constraint submanifold

Wf = W2 = w ∈ W1 : ux = uy

with the corresponding tangency conditions on h.

Proposition 3.79. Let Ω1 denote the pullback of ΩH to W1 by the natural inclusioni : W1 → W , that is Ω1 = i∗(ΩH). Suppose that dimM > 1 (resp. dimM = 1). The(m + 1)-form Ω1 is multisymplectic (resp. cosymplectic together with η) if and only if Lis regular.

Proof. First of all, assume thatm > 1 and let us make some considerations. By denition,Ω1 is multisymplectic whenever Ω1 has trivial kernel, that is,

if v ∈ TW1, ivΩ1 = 0 ⇐⇒ v = 0 .

This is equivalent to say that

if v ∈ i∗(TW1), ivΩH|i∗(TW1) = 0 ⇐⇒ v = 0 .

Let v ∈ TW be a tangent vector whose coecients in an adapted basis are given by

v = γi∂

∂xi+ Aα

∂

∂uα+ Aαi

∂

∂uαi+Bi

α

∂

∂piα+ C

∂

∂p.

Using the local expression (3.101), we may compute the contraction of ΩH by v,

ivΩH =−Biα duα ∧ dm−1xi + Aα dpiα ∧ dm−1xi − γj dpiα ∧ duα ∧ dm−2xij

+

(Aαi p

iα +Bi

αuαi − Aα

∂L

∂uα− Aαi

∂L

∂uαi

)dmx

− γj(piα duαi + uαi dpiα −

∂L

∂uαduα − ∂L

∂uαiduαi

)∧ dm−1xj.

(3.111)

In addition to this, let us consider a vector v ∈ TW tangent toW1, that is v ∈ i∗(TW1),we then have that

d

(piα −

∂L

∂uαi

)(v) = 0 and d

(p+ piαu

αi − L

)(v) = 0,

which leads us to the following relations for the coecients of v:

Biα = γj

∂2L

∂xj∂uαi+ Aβ

∂2L

∂uβ∂uαi+ Aβj

∂2L

∂uβj ∂uαi

(3.112)

C = γj∂L

∂xj+ Aα

∂L

∂uα−Bi

αuαi . (3.113)

It is important to note that, even though the coecient Aαi explicitly appears in the pre-vious equations (3.111) and (3.112), for such a vector v ∈ i∗(TW1), the terms associatedto these Aαi cancel out in the development of ivΩH, Equation (3.111). Thus, a tangentvector v ∈ i∗(TW1) is in the kernel of ΩH if and only if its coecients satisfy the followingrelations

γj = 0, Aαi = 0, Aβj∂2L

∂uβj ∂uαi

= 0, Biα = 0, C = 0.


These considerations being made, the assertion is now clear for the multidimensionalcase.

Now, let suppose m = 1 and consider coordinates (t, qα, vα, p, pα) on W , which inducecoordinates (t, qα, vα) on W1. In these coordinates, the Cartan (m+ 1)-form is written

ΩH = − dpα ∧ dqα + vα dvα ∧ dt+ pα dvα ∧ dt− dL ∧ dt

and its pull back to W1

Ω1 = − d

(∂L

∂vα

)∧ dqα + vα d

(∂L

∂vα

)∧ dt− ∂L

∂qαdqα ∧ dt.

A straightforward computation shows that

Ωn1 ∧ dt = det

(∂2L

∂vα∂vβ

)dq1 ∧ dv1 ∧ · · · ∧ dqn ∧ dvn ∧ dt,

which is a volume form if and only if L is regular.

Corollary 3.80. Under the same assumptions, we have: (J1π,ΩL), (J1π0,Ωh) and(W1,Ω1) are (globally) locally multisymplecticomorphic (resp. cosymplecticomorphic to-gether with η when m = 1) if and only if L is (hyper)regular. Indeed, W1 = graph(LegL)and the corresponding multisymplecticomorphisms (resp. cosymplecticomorphisms) are

W1pr1 |W1

||zzzzzzzz µpr2 |W1=legL pr1 |W1

""FFFFFFFF

J1π legL

// J1π0

(3.114)

In the following proposition, Wf denotes the nal constraint submanifold, which co-incides with W1 whenever L is regular.

Proposition 3.81. Let σ be a section of πWf ,M : Wf −→ M and denote σ = i σ andφ = πWf ,E σ, where i : Wf → W is the canonical inclusion. If σ is an integral sectionof h, then the Lagrangian part σ1 = pr 1 σ of σ is holonomic, i.e. σ1 = j1φ, and satisesthe Euler-Lagrange equations:

j2φ∗(∂L

∂uα− d

dxj∂L

∂uαj

)= 0. (3.115)

Proof. If σ = (xi, σα, σαi , σ0, σiα) is an integral section of h, then

∂σα

∂xj= Aαj,

∂σαi∂xj

= Aαij,∂σiα∂xj

= B iαj and

∂σ0

∂xj= Cj,

where the A's, B's and C's are the coecients given in (3.103). From Equation (3.104),we have that σ1 is holonomic, since σαi = ∂σα/∂xi. On the other hand, using the equations(3.105) and (3.106), we obtain:

0 =∂L

∂uα j1φ− ∂σjα

∂xj;

σiα =∂L

∂uαi j1φ.


We then have

0 = (j1φ)∗∂L

∂uα− (j2φ)∗

(d

dxj∂L

∂uαj

),

which is precisely the Euler-Lagrange equations.

W = J1π ×E J1π†)

PPPPPPPq

J1π J1π†

pr 2 pr 1

W ′1

W1

E

X

>

HHHHY

XXXXXX

XXXXy

/

?

HHHHH

HHHj J1π

QQQQs9

@@@@@@@@@R

π1

?

π π1

Denition 3.82. Let H be the Hamiltonian density associated to a given Lagrangiandensity L : J1π −→ ΛmM . The associated (extended) Hamiltonian action is the mapAH : ΓπW,M ×K → R given by

AH(σ,R) :=

∫R

σ∗(ΘH), (3.116)


It is called Hamilton-Pontryagin principle for eld theories in [151].

Theorem 3.83. A section σ : M → W of πW,M : W → M is a critical point of theHamiltonian action AH if and only if it satises the local equations

σαi =∂σα

∂xi,

∂σiα∂xi

=∂L

∂uα, and σiα =

∂L

∂uαi(3.117)

on M , and

L(xi, σα, σαi ) = 0 and σiα =∂L

∂uαi

∣∣∣σ(x)

= 0 (3.118)

on the boundary ∂M of M , where (xi, uα, uαi , p, piα) denotes adapted coordinates on W

and σ = (xi, σα, σαi , σ0, σiα).

Proof. As usual, given a section σ ∈ Γπ†1 and a compact region R ⊆ M , let σε =ϕε σ (ϕε)

−1 be a variation of σ such that the innitesimal generator ξ of ϕε vanishesoutside of (π†1)−1(R). The variation of the Hamiltonian action AH is then given by

d

dε[AH(σε, Rε)]

∣∣∣ε=0

=

∫R

d

dε[(ϕε σ)∗ΘH]

∣∣∣ε=0

=

∫R

σ∗(LξΘH)

= −∫R

σ∗(iξΩH) +

∫∂R∩∂M

σ∗(iξΘH).


We deduce from here that σ is a critical point of AH if and only if

σ∗(iξΩH) = 0 and σ∗(iξΘH) =∂M

0, ∀ξ ∈ X(W ).

Using local coordinates (xi, uα, uαi , p, piα) on W and denoting σ = (xi, σα, σαi , σ0, σ

iα), we

compute on the one hand

σ∗(iξΩH) =

[ξα(∂σiα∂xi− ∂L

∂uα

)+ ξαi

(σiα −

∂L

∂uαi

)+ ξiα

(σαi −

∂σα

∂xi

)+

+ ξj(∂σiα∂xj

∂σα

∂xi− ∂σiα∂xi

∂σα

∂xj− σiα

∂σαi∂xj− σαi

∂σiα∂xj

+∂L

∂uα∂σα

∂xj+

∂L

∂uαi

∂σαi∂xj

)]dmx,

and on the other hand

σ∗(iξΘH) =

[ξj(L− σiασαi ) + ξασjα + ξjσiα

∂σα

∂xj− ξiσjα

∂σα

∂xj

]dm−1xj.

From here, we conclude that, in order to be a critical point of AH, σ must satisfy theequations (3.117) and (3.118).

Note that equations in (3.117) are equivalent to equations (3.1043.106) when weconsider an integral section of a solution h of the dynamical equation (3.102). They alsocorrespond to the Euler-Lagrange equations (3.38) (combine the second and the thirdone), to the Hamilton's equations (3.77) (dene H = uαi p

iα−L and consider the rst two

equations) and the Legendre transform (3.84) (take the third equation). In the same way,the boundary conditions (3.118) are equivalent to those the Lagrangian side, Equation(3.39), and those of the Hamiltonian side, Equation (3.78) (see remarks 3.36 and 3.63).

Denition 3.84. Let L : J1π −→ ΛmM be a Lagrangian density. The associated(extended) Hamiltonian-Pontryagin action is the map AL : ΓπW,M ×K → R given by

AL(σ,R) :=

∫R

(L σ1 +

⟨σ†1, j

1σ0

⟩−⟨σ†1, σ1

⟩)(3.119)


In fact, the Hamiltonian-Pontryagin action 3.84 coincides with the Hamiltonian action3.82 as stated by the next result.

Theorem 3.85. A section σ : M → W of πW,M : W → M is a critical point of theHamiltonian-Pontryagin action AL if and only if it satises the local equations

σαi =∂σα

∂xi,

∂σiα∂xi

=∂L

∂uα, and σiα =

∂L

∂uαi(3.120)

on M , and

σiα =∂L

∂uαi

∣∣∣σ(x)

= 0 (3.121)

on the boundary ∂M of M , where (xi, uα, uαi , p, piα) denotes adapted coordinates on W

and σ = (xi, σα, σαi , σ0, σiα).


Proof. Given a section σ ∈ Γπ†1 and a compact region R ⊆M , we have that the variationof the Hamiltonian-Pontryagin action AL with respect to a variation δσ of σ is given by

δALδσ

∣∣∣∣(σ,R)

· δσ =

∫R

δ

δσ

[L(xi, σα, σαi ) + σiα

(∂σα

∂xi− σαi

)] ∣∣∣∣σ

· δσ dmx

=

∫R

[∂L

∂uαδσα +

∂L

∂uαiδσαi + δσiα

(∂σα

∂xi− σαi

)+ σiα

(∂

∂xiδσα − δσαi

)]dmx

=

∫R

[(∂L

∂uα− ∂σiα∂xi

)δσα +

(∂L

∂uαi− σiα

)δσαi +

(∂σα

∂xi− σαi

)δσiα

]dmx

+

∫∂R

σiαδσα dm−1xi.

where (xi, uα, uαi , p, piα) denotes adapted coordinates on W and σ = (xi, σα, σαi , σ0, σ

iα).

We thus deduce that σ is a critical point of AL, i.e. δAL/δσ = 0, if and only if therelations (3.120) and (3.121) are satised.

Here, the boundary conditions (3.121) dier from the boundary conditions (3.118),since in the proof we have considered vertical variations.

Chapter 4

Higher Order Classical Field Theory

In this chapter, we will nd the main original contributions of this memory. For it, we willrst extend the notions of jets to an arbitrary order, that is, higher-order jets. We willnd, as an important result, an unambiguous and intrinsic formalism for the higher-ordercalculus of variations. The case of constrained calculus will be also analyzed. The mainresults appear in [24, 25, 26, 27] and in a forthcoming paper. As a basic reference in whatfollows, the reader is refereed to the book by Saunders [139].

Through this section, (E, π,M) denotes a ber bundle whose base space M is asmooth manifold of dimension m, and whose bers have dimension n, thus E is (m+ n)-dimensional. Adapted coordinate systems in E will be of the form (xi, uα), where (xi) isa local coordinate system in M and (uα) denotes ber coordinates.

Lower case Latin (resp. Greek) letters will usually denote indexes that range between1 and m (resp. 1 and n). Capital Latin letters will usually denote multi-indexes whoselength ranges between 0 and k (see Appendix A). In particular and if nothing else it isstated, I and J will usually denote multi-indexes whose length goes from 0 to k − 1 and0 to k, respectively; and K (and sometimes R) will denote multi-indexes whose length isequal to k. The Einstein notation for repeated indexes and multi-indexes is understoodbut, for clarity, in some cases the summation for multi-indexes will be indicated.

4.1 Higher Order Jet bundles

Denition 4.1. Given a point x ∈ M , two local sections φ, ψ ∈ Γxπ are k-equivalent atx if their value coincide at x, as well as their partial derivatives up to order k

φ(x) = ψ(x) and∂kφα

∂xi1 · · · ∂xik∣∣∣x

=∂kψα

∂xi1 · · · ∂xik∣∣∣x,

for all 1 ≤ α ≤ n, 1 ≤ ij ≤ m, 1 ≤ j ≤ k. This denes an equivalence relation in Γxπ.The equivalence class containing φ is called the kth jet of φ at x and is denoted jkxφ.

The notion of k-equivalency is independent of the chosen coordinate system (adaptedor not), thus so is the equivalence relation that it denes (see [61, 64, 139], for moredetails).

Denition 4.2. The kth jet manifold of π, denoted Jkπ, is the whole collection of kthjets of arbitrary local sections of π, that is,

Jkπ :=jkxφ : x ∈M, φ ∈ Γxπ

.

61

62 CHAPTER 4. HIGHER ORDER CLASSICAL FIELD THEORY

The functions given byπk : Jkπ −→ M

jkxφ 7−→ x(4.1)

andπk,0 : Jkπ −→ E

jkxφ 7−→ φ(x)(4.2)

are called the kth source projection and the kth target projection respectively.

From the denitions, it is trivial to see that j0xφ = φ(x), J0π = E, π0 = π and

π0,0 = IdE.

Proposition 4.3. The kth jet manifold of π, Jkπ, may be endowed with a structure ofsmooth manifold. A system of adapted coordinates (xi, uα) on E induces a system ofcoordinates (xi, uαI ) (with 0 ≤ |I| ≤ k) on J1π such that

xi(jkxφ) = xi(x) and uαI (jkxφ) =∂|I|φα

∂xI

∣∣∣x.

In the induced local coordinates (xi, uαI ), the source and the target projections arewritten

πk(xi, uαI ) = (xi) and πk,0(xi, uαI ) = (xi, uα). (4.3)

From here, it is clear that πk and πk,0 are certainly projections (surjective submersions)over M and E, respectively. Therefore, (Jkπ, πk,M) and (Jkπ, πk,0, E) are ber bundles.

If we consider a change of coordinates (xi, uα) 7→ (yj, vβ) in E, it induces a changeof coordinates (xi, uαI ) 7→ (yj, vβJ ) in J1π. In this case, the velocities transform by thefollowing rule:

vβJ+1j=

(∂vβJ∂xi

+ uαI+1i

∂vβJ∂uαI

)∂xi

∂yj(4.4)

=∂vβJ∂xi

∂xi

∂yj+ uαI′

∑I+1i=I′

∂vβJ∂uαI

∂xi

∂yj,

from where we deduce that coordinates of a particular order depend only on coordinatesof equal or lower order, that is

vβJ = vβJ (xi, uαI ) : |I| ≤ |J |.

Even more, the changes have a polynomial expansion and it is ane from order to order(cf. [139]).

Proposition 4.4. For each 0 ≤ l ≤ k, dene the map

πk,l : Jkπ −→ J lπjkxφ 7−→ jlxφ.

(4.5)

We have that (Jkπ, πk,l, Jlπ) are smooth ber bundles to which the induced coordinates

(xi, uαI ) are adapted. Moreover, for the particular case l = k − 1, (Jkπ, πk,k−1, Jk−1π) is

an ane bundle, being its associated vector bundle

π∗k−1(SkT ∗M)⊗Jk−1π π∗k−1,0(V π),

where SkT ∗M is the space of symmetric covariant tensors of order k over M and V π thevertical bundle of π.

4.1. HIGHER ORDER JET BUNDLES 63

Jkππk,k−1 //

πk

))SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS · · · π3,2 // J2ππ2,1 //

π2

##HHHHHHHHHHHHHHHHHHH J1ππ1,0 //

π1

4444444444444 E

π

M

φ

YY

jkφ

``

Figure 4.1: Chain of jets

In the induced local coordinates (xi, uαI ) of Jkπ, with 0 ≤ |I| ≤ k, and (xi, uαJ) of J lπ,with 0 ≤ |J | ≤ l ≤ k, we have the obvious local expression

πk,l(xi, uαI ) = (xi, uαJ).

4.1.1 Prolongations, lifts and contact

Denition 4.5. Let φ ∈ Γπ be a (local) section, its kth prolongation is the (local) sectionof πk,0 given by

(jkφ)(x) := jkxφ,

for every x ∈ M . An arbitrary (local) section σ of πk is said to be holonomic if it is thekth prolongation of a (local) section φ ∈ Γπ, that is, if σ = jkφ.

Denition 4.6. Let f : E → F be a bundle morphism between two ber bundles(E, π,M) and (F, ρ,N), such that the induced function on the base, f : M → N , is adieomorphism. The kth prolongation of f is the map jkf : Jkπ → Jkρ given by

(jkf)(jkxφ) := jkf(x)

φf , ∀jkxφ ∈ Jkπ,

where φf := f φ f−1 ∈ Γρ.

Jkπjkf //

πk,0

Jkρ

ρk,0

E

f //

π

F

ρ

M

f //

φ

HHjkφ

FF

N

φf

VVjkφf

XX

Figure 4.2: The kth prolongation of a morphism

Note that the kth prolongation jkf of a morphism f is not only a morphism between(Jkπ, πk,0, E) and (Jkρ, ρk,0, F ), and a morphism between (Jkπ, πk,M) and (Jkρ, ρk, N),but also a morphism between the intermediate lth jet bundles (Jkπ, πk,l, J

lπ) and (Jkρ,


ρk,l, Jlρ), for 0 < l < k. In each case, the induced functions between the corresponding

base spaces are f , f and jlf , respectively.If (xi, uαI ) and (yj, vβ, vβJ ) denote adapted coordinates in Jkπ and Jkρ, respectively,

then we have

fβJ+1j= vβJ+1j

jkf =

(∂fβJ∂xi

+ uαI+1i

∂fβJ∂uαI

)· ∂f

−i

∂yj.

The expression between brackets is called the total derivative of fβJ with respect to xi.We will come back to it later.

Denition 4.7. Let φ : M → E be a section of π, x ∈ M and u = jk−1x φ. The vertical

dierential of the section φ at the point u ∈ Jk−1π is the map

dvuφ : TuJ

k−1π −→ Vu πk−1

v 7−→ v − Tu(jk−1φ πk−1)(v)

Namely, dvuφ := Idu−Tu(jk−1φ πk−1).

Notice that the image of dvuφ is certainly in Vu πk−1 since Tuπk−1 dv

uφ = 0 and that,in fact, dv

uφ depends only on jkxφ. In adapted local coordinates (xi, uαI ) of Jk−1π,

dvuφ =

(duαI −

∂|I|+1φα

∂xI+1idxi)⊗ ∂

∂uαI. (4.6)

Denition 4.8. The canonical structure form of Jkπ is the 1-form θ on Jkπ with valuesin V πk−1 dened by

θjkxφ(V ) := ( dvjk−1x φ

φ)(Tjkxφπk,k−1(V )), V ∈ TjkxφJkπ, (4.7)

where φ is any representative of jkxφ ∈ Jkπ. The contraction of the covectors in V∗ πk−1

with θ denes a distribution in T ∗Jkπ. This distribution is called the contact moduleor the Cartan codistribution (of order k) and it is denoted Ck. Its elements are contactforms. The annihilator of Ck is the Cartan distribution (of order k).

Note that the expression (4.7) does not depend on the representative φ of jkxφ, henceit is well dened. In adapted local coordinates (xi, uαI , u

αK) of Jkπ, where 0 ≤ |I| ≤ k− 1

and |K| = k,

θ =(

duαI − uαI+1idxi)⊗ ∂

∂uαI. (4.8)

In fact, the contact forms duαI − uαI+1idxi ∈ Ck are a base of the contact module.

Proposition 4.9. Let (xi, uαI , uαK) be adapted coordinates on Jkπ, where 0 ≤ |I| ≤ k − 1

and |K| = k, a basis of the Cartan codistribution is given by the coordinate contact forms

θαI = duαI − uαI+1idxi. (4.9)

Proposition 4.10. The canonical structure form θ ∈ Γ(T ∗Jkπ⊗Jkπ V π) and the contactforms ω ∈ Ck are pulled back to zero by the kth prolongation jkφ of any section φ of π.Moreover, this characterizes the module of contact forms, i.e.

ω ∈ Ck ⇔ (jkφ)∗ω = 0, ∀φ ∈ Γπ. (4.10)


Proof. Let ω ∈ Ω(Jkπ) be an arbitrary form. We can write ω as the linear combination

ω = ωi dxi + ωJα duαJ , 0 ≤ |J | ≤ k,

where the ω's are unknown functions on Jkπ. Given any section φ of π, we have that

(jkφ)∗ω =

(ωi jkφ+ (ωIα jkφ) · ∂

|I|+1φα

∂xI+1i+ (ωKα jkφ) · ∂

k+1φα

∂xK+1i

)dxi = 0.

Since two k-equivalent sections at a point x ∈ M coincide on their partial derivatives atx up to order k, we deduce that

ωKα = 0 and ωi + ωIαuαI+1i

= 0.

Substituting ωi and ωKα in the initial expression of ω, we obtain

ω = −ωIαuαI+1idxi + ωJα duαJ = ωIα( duαI − uαI+1i

dxi) = ωJαθαI ,

which proofs the suciency by Proposition 4.9.The necessity is immediate.

A complementary or dual result to the previous one is the following.

Proposition 4.11. Let σ ∈ Γπk be a (local) section. The following statements areequivalent:

1. σ is holonomic.

2. σ pulls back to zero any contact form, that is

σ∗ω = 0, ∀ω ∈ Ck. (4.11)

Notice that the contact forms are πk,k−1-basic, which is clear from the coordinateexpression (4.9). Though, therefore they may be thought as forms along πk,k−1 ratherthan on Jkπ. In this sense are dened total derivatives.

Denition 4.12. A total derivative is a vector eld ξ along πk,k−1 which is annihilated bythe Cartan codistribution (as forms along πk,k−1). Given a system of adapted coordinates(xi, uα, uαI , u

αK) in Jkπ, where 0 ≤ |I| ≤ k − 1 and |K| = k, the local vector elds dened

along π1,0 byd

dxi=

∂

∂xi+ uαI+1i

∂

∂uα(4.12)

are called coordinate total derivatives.

It is immediate to check that coordinate total derivatives are total derivatives, in factthey dene a basis of such vector elds. Under a change of coordinates, (xi, uα) to (yj, vβ),a coordinate total derivative transforms linearly by the Jacobian of the underlying changeof coordinates:

d

dyj=∂xi

∂yjd

dxi.


If ξ ∈ X(πk,k−1) has the dierent coordinate representations

ξ = ξid

dxi= ξj

d

dyj,

where the coecients ξi and ξj are functions on Jkπ. Then,

ξi = ξj∂xi

∂yj.

Denition 4.13. The total lift of a vector eld ξ = ξi∂i on M is the unique totalderivative that projects on ξ itself, that is, the vector eld ξk along πk,k−1 locally givenby

ξk(jkxφ) = ξi(x)d

dxi

∣∣∣jkxφ.

Note that the total lift of the coordinate partial derivatives in M are precisely thecoordinate total derivatives.

Now, consider the action of total derivatives on smooth functions over Jk−1π. Iff ∈ C∞(Jk−1π), the action of d/ dxi on it yields a function df/ dxi ∈ C∞(Jkπ). Inparticular, the action of d/ dxi on the coordinate function uα ∈ C∞(E), gives as expected

duαIdxi

= uαI+1i∈ C∞(Jkπ), ∀ 0 ≤ |I| ≤ k − 1.

Another interesting fact is how total derivatives and jets are related. Let f ∈ C∞(J lπ),l < k, φ ∈ Γπ and ξ ∈ X(M), we have

ξ(f jlφ) = ξk(f) jl+1φ,

in coordinates∂(f φ)

∂xi=

df

dxi jkφ. (4.13)

Finally, note that coordinate total derivatives and ordinary partial derivates do not nece-sarilly conmute:

∂

∂xidf

dxj=

d

dxj∂f

∂xi,

∂

∂uαdf

dxj=

d

dxj∂f

∂uαbut

∂

∂uαJ

df

dxi=

d

dxi∂f

∂uαJ+ δJI+1i

∂f

∂uαI,

where f ∈ C∞(E). Nevertheless, coordinate total derivatives do commute, what allow usto use the multi-index notation with iterated coordinate total derivatives.

Proposition 4.14. Let f ∈ C∞(J lπ), then dfdxi∈ C∞(J l+1π) and d

dxjdfdxi∈ C∞(J l+2π).

Moreover, we have thatd

dxjdf

dxi=

d

dxidf

dxj.

Denition 4.15. Given a vector eld ξ on E, its kth lift (or kth jet) is the unique vectoreld ξ(k) on Jkπ that is projectable to ξ by πk,0 and preserves the Cartan codistributionwith respect to the Lie derivative.


Proposition 4.16. Let ξ be a vector eld on E. If ξ has the local expression

ξ = ξi∂

∂xi+ ξα

∂

∂uα(4.14)

in adapted coordinates (xi, uα) on E, then its kth-lift ξ(k) has the form

ξ(k) = ξi∂

∂xi+ ξαJ

∂

∂uαJ(4.15)

for the induced coordinates (xi, uαJ) on Jkπ, where

ξα0 = ξα and ξαI+1i=

dξαIdxi− uαI+1j

dξj

dxi. (4.16)

In particular, if ξ is vertical with respect to π, then ξαJ = d|J |ξα/ dxJ .

Proof. Since ξ(k) is πk,0-projectable to ξ, it must have the form

ξ(k) = ξi∂

∂xi+ ξαJ

∂

∂uαJ

where ξα0 = ξα and where the remaining components ξαJ , with |J | = 1, . . . , k, still have tobe determined.

Note that the preserving condition is equivalent to require that the Lie derivativesby ξ(k) of the elements of any xed base of the Cartan codistribution Ck are still contactforms. Thus, consider the base θαI given in Proposition (4.9) and let us compute theLie derivative of its elements by ξ(k). Using the Cartan's formula L = d i + i d, weobtain

Lξ(k)θαI = Lξ(k)( duαI − uαI+1idxi)

=∂ξαI∂xi

dxi +∂ξαI

∂uβJduβJ − u

αI+1i

∂ξi

∂xjdxj − uαI+1i

∂ξi

∂uβduβ − ξαI+1i

dxi

Adding and subtracting properly some terms, we have

Lξ(k)θαI =∂ξαI∂xi

dxi +∂ξαI

∂uβI

θβI

+∂ξαI

∂uβI

uβI+1i

dxi +∂ξαI

∂uβKduβK

−uαI+1i

∂ξi

∂xjdxj − uαI+1i

∂ξi

∂uβθβ − uαI+1i

∂ξi

∂uβuβj dxj − ξαI+1i

dxi.

As Lξ(k)θαI is required to be contact,

ξαI+1i=∂ξαI∂xi

+ uβI+1i

∂ξαI

∂uβI

− uαI+1j

(∂ξj

∂xi+ uβi

∂ξj

∂uβ

)and

∂ξαI

∂uβK= 0.

From the rst equation we deduce that ξαJ depends only on uαI 's with |I| ≤ |J |, whichagrees with the second one. Rewriting the former in terms of the coordinate total deriva-tives (4.12), we nally obtain

ξαI+1i=

dξαIdxi− uαI+1j

dξj

dxi.

The nal statement is clear from here.


Corollary 4.17. Under the same assumptions, we have that the components of the kthlift ξ(k) ∈ X(Jkπ) of a vector eld ξ ∈ X(E) are explicitly given by

ξαJ =d|J |ξα

dxJ−

∑Iu+Iξ=J|Iu|,|Iξ|6=0

J !

Iu!Iξ!uαIu+1l

d|Iξ|ξl

dxIξ− uαl

d|J |ξl

dxJ. (4.17)

Proof. We proceed by induction on the length |J | of a multi-index J ∈ Nm. For J = 1j,with 1 ≤ j ≤ m, we obtain

ξαj =dξα

dxj− uαl

dξl

dxj,

which agrees with the recursive formula (4.15) (and also with (3.18)). Let us assume thatthe theorem is true for multi-indexes up to length k − 1 ≥ 1 and consider a multi-indexK ∈ Nm of length k. For any decomposition K = J + 1j, where J ∈ Nm and 1 ≤ j ≤ m,we have

ξαJ+1j=

dξαJdxj− uαJ+1l

dξl

dxj

=d|J |+1ξα

dxJ+1j−

∑Iu+Iξ=J|Iu|,|Iξ|6=0

J !

Iu!Iξ!

(uαIu+1j+1l

d|Iξ|ξl

dxIξ+ uαIu+1l

d|Iξ|+1ξl

dxIξ+1j

)

−uα1j+1l

d|J |ξl

dxJ− uαl

d|J |+1ξl

dxJ+1j− uαJ+1l

dξl

dxj,

where we have used the formula (4.15) and the induction hypothesis. We multiply eachmember of the equality by K(j)/|K| and sum over all the decompositions of the typeK = J + 1j, what gives us thanks to Lemma A.4

ξαK =d|K|ξα

dxK−

∑J+1j=K

K(j)

|K|∑

Iu+Iξ=J|Iu|,|Iξ|6=0

J !

Iu!Iξ!

(uαIu+1j+1l

d|Iξ|ξl

dxIξ+ uαIu+1l

d|Iξ|+1ξl

dxIξ+1j

)

−∑

J+1j=K

K(j)

|K|

(uα1j+1l

d|J |ξl

dxJ+ uαJ+1l

dξl

dxj

)− uαl

d|K|ξl

dxK

=d|K|ξα

dxK−

∑Iu+Iξ+1j=K|Iu|,|Iξ|6=0

K(j)

|K|(Iu + Iξ)!

Iu!Iξ!

(uαIu+1j+1l

d|Iξ|ξl

dxIξ+ uαIu+1l

d|Iξ|+1ξl

dxIξ+1j

)

−∑

J+1j=K

K(j)

|K|

(uα1j+1l

d|J |ξl

dxJ+ uαJ+1l

dξl

dxj

)− uαl

d|K|ξl

dxK.

We now need to rearrange properly the middle terms. For the rst one, we substitute


Iu + 1j by Ju and Iξ by Jξ, obtaining

∑Iu+Iξ+1j=K|Iu|,|Iξ|6=0

K(j)

|K|(Iu + Iξ)!

Iu!Iξ!uαIu+1j+1l

d|Iξ|ξl

dxIξ=

=∑

Ju+Jξ=K|Iu|≥2,|Jξ|6=0

∑Iu+1j=Ju

Ju(j)

|K|K!

Ju!Jξ!uαJu+1l

d|Jξ|ξl

dxJξ

=∑


|Ju||K|

K!

Ju!Jξ!uαJu+1l

d|Jξ|ξl

dxJξ,

where we have use the fact that K(j)(Iu + Iξ)! = K! and Ju(j)Iu! = Ju! (Lemma A.1)and again the identity (A.7). For the third middle term, we substitute Iu + 1j by Ju andIξ by Jξ, obtaining

∑J+1j=K

K(j)

|K|uα1j+1l

d|J |ξl

dxJ=

∑Ju+Jξ=K|Iu|=1,|Jξ|6=0

|Ju||K|

K!

Ju!Jξ!uαJu+1l

d|Jξ|ξl

dxJξ

where we have use the fact that K(j)Jξ! = K! and |Ju| = Ju! = 1. The second and forthmiddle terms are rearranged accordingly. We thus arrive to

ξαK =d|K|ξα

dxK− uαl

d|K|ξl

dxK

−∑


|Ju||K|

K!

Ju!Jξ!uαJu+1l

d|Jξ|ξl

dxJξ−

∑Ju+Jξ=K|Iu|=1,|Jξ|6=0

|Ju||K|

K!

Ju!Jξ!uαJu+1l

d|Jξ|ξl

dxJξ

−∑

Ju+Jξ=K|Iu|6=0,|Jξ|≥2

|Jξ||K|

K!

Ju!Jξ!uαJu+1l

d|Jξ|ξl

dxJξ−

∑Ju+Jξ=K|Iu|6=0,|Jξ|=1

|Jξ||K|

K!

Ju!Jξ!uαJu+1l

d|Jξ|ξl

dxJξ

=d|K|ξα

dxK−

∑Ju+Jξ=K|Ju|,|Jξ|6=0

|Ju|+ |Jξ||K|

K!

Ju!Jξ!uαJu+1l

d|Jξ|ξl

dxJξ− uαl

d|K|ξl

dxK,

which is the desired formula since |Ju|+ |Jξ| = |K|.

Originally, the kth lift is dened for π-projectable vector elds on E. The kth lift ofsuch vector eld ξ is the innitesimal generator of the kth lift of the ow of ξ. Denition4.15 is a characterization of this property and it is generalized for any kind of vector eldson E (see [71]).

Proposition 4.18. Let ψε be the ow of a given π-projectable vector eld ξ over E.Then, the ow of ξ(k) is the kth prolongation of ψε, jkψε.


4.1.2 On the denition of vertical endomorphisms

We are going to face one of the rst problems in order to dene a canonical geometricLagrangian formalism. There is no natural extension of the notions of vertical endomor-phism for rst order theories (see Section 3.1.2). However, an alternative approach wasdeveloped by Saunders in [138].

Denition 4.19. Given a k-jet jkxφ ∈ Jkπ, let A ∈ SkT ∗xM ⊗jkxφ Vφ(x) π. The vertical liftof A at jkxφ is the tangent vector Av

jkxφ∈ Tjkxφ(Jkπ) given by

Avjkxφ

(f) =d

dtf(jkxφ+ tA)

∣∣t=0, ∀f ∈ C∞(Jkjkxφπ). (4.18)

By the very denition of vertical lift, given a smooth function f ∈ C∞(Jk−1π),

(Tjkxφπk,k−1)(Avjkxφ

)(f) = Avjkxφ

(f πk,k−1)

=d

dt(f πk,k−1)(jkxφ+ tA)

∣∣t=0

=d

dtf(φ(x))

∣∣t=0

= 0.

Thus, the vertical lift takes values into the vertical ber bundle V πk,k−1 ⊂ TJkπ. Indeed,it is a morphism of vector bundles over the identity of Jkπ,

(·)v : SkT ∗M ⊗Jkπ V π −→ V πk,k−1.

Note that, this time, the tensor product is taken over Jkπ and not over E. Note also thatfor each jkxφ ∈ Jkπ, the vertical lift at jkxφ,

(·)vjkxφ

: SkT ∗xM ⊗ Vφ(x) π −→ Vjkxφ πk,k−1 ⊂ TjkxφJkπ,

is a linear isomorphism. In adapted local coordinates (xi, uαJ), if A = AαK dxK |x ⊗∂/∂uα|φ(x), where dxK is the symmetric tensor product of the local 1-forms dxi1 , . . . ,dxik with K = 1i1 + · · ·+ 1ik , then

Avjkxφ

= AαK∂

∂uαK

∣∣∣jkxφ

and (·)v = duα ⊗ δ

δxK⊗ ∂

∂uαK, (4.19)

where δ/δxK is the dual counterpart of dxK .Now, we would like to use this vertical lift in order to generalize the denitions of

the vertical endomorphisms of rst order, denitions 3.18 and 3.19. Nevertheless, theideas that are behind these denitions seem to not work for this one. In the case of thevolume dependent vertical endomorphism 3.18, one would like to dene a skewsymmetricmap Sη : (TJkπ)m → TJkπ using the volume form η and the vertical lift (·)v, butthere is no chance to obtain an element in the domain of (·)v from a tangent vectorin TJkπ. In the case of the canonical vertical endomorphism 3.19, we look for a mapS : T ∗M ⊗ TJkπ → V πk,0, but the contraction of the canonical form η with the verticallift (·)v simply does not give what one would expect.


Therefore, we are forced to try to generalize these objects my means of their localdescriptions, equations (3.23) and (3.24). For instance, the obvious formula for a canonicalvertical endomorphism for higher-order jet bundles would be

Sk :=k−1∑|I|=0

θαI ⊗∂

∂xi⊗ ∂

∂uαI+1i

. (4.20)

Unfortunately, this local denition does not behave as expected under a change of coor-dinates. Let (xi, uαI ) and (yj, vβJ ) denote two systems of adapted coordinates in Jkπ then,following the transformation rules (4.4), we have

θαI ⊗∂

∂xi⊗ ∂

∂uαI+1i

=∑

|I|+1≤|J |≤k

∂uαI

∂vβJ· ∂y

j

∂xi·∂vβ

J

∂uαI+1i

· θβJ ⊗∂

∂yj⊗ ∂

∂vβJ

,

where we have omitted some of the summation symbols for clarity. For the second ordercase, after further computations, this is translated to

θα ⊗ ∂

∂xi⊗ ∂

∂uα+ θαi′ ⊗

∂

∂xi⊗ ∂

∂uαi′i=

= θβ ⊗ ∂

∂yj⊗ ∂

∂vβj+ θβj′ ⊗

∂

∂yj⊗ ∂

∂vβj′j

+

(∂2uα

∂yj′∂vβ+ vβ

′

j′∂2uα

∂vβ′∂vβ

)· ∂v

β

∂uαθβj′ ⊗

∂

∂yj⊗ ∂

∂vβj′j

.

Obviously, this is not invariant under a change of coordinates.

4.1.3 Partial dierential equations

Lemma 4.20. If N is an open submanifold of M , then Jk(πN) ' π−1k (N).

Denition 4.21. A dierential equation on π is a closed embedded submanifold Pof the jet manifold Jkπ. The order of P is the largest natural number r satisfyingπ−1r,r−1(πk,r−1) 6= πk,rP . A solution of P is a local section φ ∈ ΓNπ, where N is an open

submanifold of M , which satises jkxφ ∈ P for every x ∈ N . A dierential equation P issaid to be integrable at z ∈ P if there is a solution φ of P (around some neighborhoodN of πk(z)) such that z = jkπk(z)φ. A rst-order dierential equation P is said to beintegrable in a subset P ′ ⊂ P if it is integrable at each z ∈ S. A rst-order dierentialequation P is said to be integrable if it is integrable at each z ∈ P .

If l is the codimension of P (dim Jkπ − dimP), there locally exist submersions Ψ :Jkπ → Rl for whom P is the zero level set. Written in local coordinates, P is given bythe set of points that satisfy

Ψµ(xi, uαJ) = 0, µ = 1, . . . , l.

Thus, dierential equations are a geometric interpretation of the usual kth-order partialdierential equations. Under certain conditions (for instance, if πk,k−1|P : P → Jk−1π is


a surjective submersion), one could solve the previous equation for some of the highest-order velocities uαK making them to depend on the other variables. For simplicity, if n = 1and we x l multi-indexes K of length k, which we denote with a hat K, the previousequation could be equivalent to the following expression

uK = φK(xi, uI , uK),

where the multi-index with check accent, K, is a multi-index of length k complementaryto those of K. For instance, in the equation

uxy = uy · uxx + ux · uyy

dened on J2π where π = pr 1 : R2 × R→ R2 with global coordinates (x, y, u), the hatmulti-index would be xy = 1x + 1y and the check ones xx = 1x + 1x and yy = 1y + 1y.

In what follows, constrained coordinates will be denoted generically with a hat accent ˆ , while free coordinates will be denoted with a check accent ˇ , i.e. uα

Kand uα

K. Note

that in general, α and α or K and K may coincide, what do not are the pairs (α, K) and(α, K).

Remark 4.22. As our ultimate goal is to characterize holonomic jet sections that belongto P , one could look for a submanifold P ′ of P consisting of the image of such sections.The submanifold P ′ is given by the constraint functions of P plus their consequences upto order k, that is, Ψµ, dΨµ

dxi, d2Ψµ

dxij, etc. Geometrically, P ′ is obtained as the output of the

following recursive process:

P(s,r) :=

P , s = 0, r = k;

πk,0(P(s−1,k)), s > 0, r = 0;J1P(s,r−1) ∩ πk,r(P(s−1,k)), s > 0, 0 < r ≤ k;

(4.21)

which stops when, for some step s ≥ 0, P(s+1,k) = P(s,k). This algorithm is a generalizationto jet bundles of the method given in [127] by Mendella et al. to extract the integral partof a dierential equation in a tangent bundle. The reader is also refereed to the alternativeapproach by Gasqui [90].

For instance, if one considers the null divergence restriction ux + vy = 0 in the 2nd-order jet manifold of pr 1 : R3 × R2 −→ R3, then the resulting manifold P(2,2) = P(1,2) isgiven by the restrictions ux + vy = 0, uxt + vyt = 0, uxx + vxy = 0 and uxy + vyy = 0 (seeExample 4.61). Note that in this particular example, the original equation is in essencea 1st-order dierential equation while, after the recurrence algorithm, it 2nd-order onesince it has been considered in J2 pr 1.

4.1.4 Iterated jet bundles

In Section 3.1, we already saw that J1π is a ber bundle over M . We thus may considerthe rst jet bundle J1π1 of the rst source projection π1 : J1π →M . Bearing this idea inmind, we could even consider arbitrary iteration of jet bundles of any order. Of greaterimportance are the jet bundles which are the rst jet of a (k − 1)th source projections:J1πk−1.

Denition 4.23. Let k, l ≥ 0, the (l, k)-iterated jet bundle of π is the lth jet bundle J lπkof the kth source projection πk : Jkπ →M .


If (xi, uα) are adapted coordinates on E and (xi, uαI ), 0 ≤ |I| ≤ k, are the correspond-ing induced local coordinates on Jkπ, adapted coordinates on the iterated jet bundle J lπkwill have the form

(xi, uαI;J), where 0 ≤ |I| ≤ k, 0 ≤ |J | ≤ l,

such that, for any local section ψ : M → Jkπ of πk,

uαI;J(jlxψ) =∂|J |ψαI∂xJ

∣∣∣∣x

,

being ψαI = uαI ψ.Let φ : M → E be a local section of π around x ∈ M , then its kth prolongation jkφ

is a local section of πk and, hence, its lth jet at x, jlx(jkφ), is and element of the iterated

jet bundle J lπk. Besides, jk+lx φ is an element of the higher order jet bundle Jk+lπ. In

fact, Jk+lπ is naturally embedded in J lπk.

Denition 4.24. The map il,k : Jk+lπ → J lπk is dened by

il,k(jk+lx φ) = jlx(j

kφ).

The elements in the image of il,k are called holonomic.

Do not confuse this concept with the one given in Denition 4.5, even though they arerelated. A holonomic iterated jet jlxσ ∈ J lπk (in the sense of 4.24), is the jet of a holonomicjet σ = jkφ (in the sense of 4.5) or, the iterated jet of a xed section jlxσ = jlx(j

kφ).In adapted coordinates (xi, uαI;J) on J lπk and (xi, uαK) on Jk+lπ, where 0 ≤ |I| ≤ k,

0 ≤ |J | ≤ l and 0 ≤ |K| ≤ k + l,

uαI;J(il,k(jk+lx φ)) = uαI+J .

It follows that Jk+lπ may be seen as the submanifold of holonomic jets of J lπk given bythe coordinate expression

Jk+lπ =jlxψ ∈ J lπk : uαI1;J1

(jlxψ) = uαI2;J2(jlxψ) whenever I1 + J1 = I2 + J2

.

J lπk

πk+l,l

yyttttttttttttttttttttt

il,k

πk+l,k

%%JJJJJJJJJJJJJJJJJJJJJ

J lπ

πl

%%KKKKKKKKKKKKKKKKKKKKK J lπkjlπk,0oo

(πk)l

(πk)l,0 // Jkπ

πk

yyssssssssssssssssssssss

M

Figure 4.3: Iterated jets


Do not confuse this notion of holonomy with the one given in Denition 4.1.1. Theformer refers to holonomy as an iterated jet, the latter as a jet section by itself.

In contrast to the elements of il,k(Jk+lπ), that are called holonomic, the elements ofJ lπk are sometimes refereed as non-holonomic jets, even though the holonomic jets belongto it. But there are still a set of particular interest between them when l = 1 (see [139]).

Two dierent maps may be dened from J1πk to J1πk−1. First, the composition of thetarget projection (πk)1,0 : J1πk → Jkπ with the natural embedding i1,k−1 : Jkπ → J1πk−1.And secondly, the rst prolongation j1πk,k−1 of πk,k−1 : Jkπ → Jk−1π as a morphism overthe identity on M . Finally, we recall that the vector bundle associated to the anebundle (πk−1)1,0 : J1πk−1 → Jk−1π is

(τJk−1π|V πk−1) pr 2 : T ∗M ⊗Jk−1π V πk−1 −→ Jk−1π.

Denition 4.25. The k-jet Spencer operator is the map

Dk : J1πk −→ T ∗M ⊗Jk−1π V πk−1

such that Dk(j1xψ) is the unique element of T ∗M ⊗Jk−1π V πk−1 whose ane action on

J1πk−1 maps (i1,k−1 (πk)1,0)(j1xψ) to (j1πk,k−1)(j1

xψ)

In local coordinates, the k-jet Spencer operator has the expression

Dk(xi, uαJ , u

αI;i) =

k−1∑|I|=0

(uαI;i − uαI+1i) dxi ⊗ ∂

∂uαI. (4.22)

Denition 4.26. The semi-holonomic (k + 1)-jet manifold Jk+1π is the submanifoldD−1k (0) of J1πk.

From the local expression of the k-jet Spencer operator, it follows that

Jk+1π =j1xψ ∈ J1πk : uαI;i(j

1xψ) = uαI+1i

(j1xψ) when 0 ≤ |I| ≤ k

.

We now have the inclusions ii,k(Jk+1) ⊂ Jk+1π ⊂ J1πk. In terms of coordinates, we maysay that the semi-holonomic manifold Jk+1π is the collection of elements of J1πk whosecoordinates are symmetric with respect to the multi-indexes up to order k, whereas theholonomic manifold Jk+1π is the collection of elements of Jk+1π whose coordinates arein addition symmetric with respect to the multi-indexes of order k + 1. We may take(xi, uαJ , u

αK;i) as coordinates of J

k+1π, where 0 ≤ |J | ≤ k and |K| = k.

4.1.5 The kth Dual Jet Bundle

There are mainly two possible choices to dene the dual space of Jkπ. For our purposes,one of them is not valid, while the other will introduce some problems in the formulationof dynamics. In spite of it all, we shall show the reason of this election.

Recall that πk,k−1 : Jkπ → Jk−1π is an ane bundle (Proposition 4.4). Thus, we mayconsider its ane dual

A :=⋃

u∈Jk−1π

Aff(Jkuπ,Λmπk−1(u)M).


Note that, in this case, the ane nature of this space takes only in consideration thehighest order component of Jkπ, what is clearer when one considers local coordinates.Let (xi, uαJ) denote adapted coordinates on Jkπ, where 0 ≤ |J | ≤ k, then they inducecoordinates (xi, uαI , p

k, pKα ) in A, where 0 ≤ |I| ≤ k − 1 and |K| = k. Note that there isonly one coordinate pk, with little k. The pairing will then be

pk + pKα uαK .

Roughly speaking, this space has the nice property of having as many momenta (plusone) as highest order velocities has Jkπ. Nonetheless, the lack of taking care of the lowerorder velocities is too important to neglect it.

A workaround could be to consider ber products of this space for each level J lπfrom l = 1 to l = k. Then we would have a space whose coordinates will take the form(xi, uαI , p

1, . . . , pk, pJα), where 0 ≤ |I| ≤ k − 1 and 1 ≤ |J | ≤ k. The problem now isthat there are many ane components pl, which would give a lack of unicity when theHamiltonian formalism would be introduced. Moreover, there is not a canonically denepairing since there are pairings dened at each level but not globally.

The alternative to all of this is to consider the iterated jet J1πk−1 and its dual spaceas ane bundle over Jk−1π. As already seen, Jkπ is anely embedded into J1πk−1, thusit makes sense to restrict the elements of J1(πk−1)† to Jkπ.

Denition 4.27. The kth dual jet bundle of π, denoted Jkπ†, is the reunion of the anemaps from J1

uπk−1 to Λmπk−1(u)M , where u is an arbitrary point of Jk−1π. Namely,

Jkπ† := J1(πk−1)† =⋃

u∈Jk−1π

Aff(J1uπk−1,Λ

mπk−1(u)M). (4.23)

The functions given byπ†k : Jkπ† −→ Mω ∈ Jkuπ† 7−→ πk−1(u)

(4.24)

andπ†k,0 : Jkπ† −→ E

ω ∈ Jkuπ† 7−→ πk−1,0(u)(4.25)

where Jkuπ† = Aff(Jkuπ,Λ

mπ(u)M) , are called the kth dual source projection and the kth

dual target projection respectively. Finally, we denote π†k,k−1 the map

π†k,k−1 : Jkπ† −→ Jk−1π

ω ∈ Jkuπ† 7−→ u(4.26)

and π†k,l := π†k,k−1 πk−1,l, for 0 ≤ l ≤ k − 1.

The duality nature of Jkπ† gives rise to a natural pairing between its elements andthose of Jkπ. The pairing will be denoted by the usual angular brackets, 〈 , 〉 : Jkπ†⊗Jk−1π

Jkπ → ΛmM .

Proposition 4.28. The kth dual jet bundle of π, Jkπ†, may be endowed with a structureof smooth manifold. A system of adapted coordinates (xi, uα) in E induces a system ofcoordinates (xi, uαI , p, p

Iiα ) in Jkπ†, where 0 ≤ |I| ≤ k−1, such that, for any jkxφ ∈ Jkπ and

any ω ∈ Jkjk−1x φ

π†, xi(ω) = xi(x), uαI (ω) = uαI (jk−1x φ) and

⟨ω, jkxφ

⟩= (p+ pIiα u

αI+ii

) dmx.


In the induced local coordinates (xi, uαI , p, pIiα ), the kth dual source and target projec-

tions are respectively written

π†k(xi, uαI , p, p

Iiα ) = (xi) and π†k,0(xi, uαI , p, p

Iiα ) = (xi, uα), (4.27)

and for the intermediate projections

π†k,l(xi, uαI , p, p

Iiα ) = (xi, uαJ), (4.28)

where 1 ≤ |J | ≤ l ≤ k−1. From here, it is clear that π†k and π†1,0 are certainly projections

overM and E respectively. Therefore, (Jkπ†, π†k,M) and (Jkπ†, π†k,0, E) are ber bundles.If we consider a change of coordinates (xi, uα) 7→ (yj, vβ) in E, it induces a change ofcoordinates (xi, uαI , p, p

Iiα ) 7→ (yj, vβI , q, q

Jjβ ) in Jkπ†. In this case, the momenta transform

by the following rule.

Proposition 4.29. Let (xi, uα) and (yj, vβ) be adapted coordinates on E and let (xi, uαI ,p, pIiα ) and (yj, vβJ , q, q

Jjβ ) be the corresponding induced coordinates on the space of semi-

basic forms Λm2 J

k−1π, where 0 ≤ |I|, |J | ≤ k. We have that the ber coordinates (withrespect to Jk−1π) transform according to the following rule:

p = Jac(y(x))

(∂vβJ∂xi

qJjβ∂xi

∂yj+ q

), (4.29)

pIiα = Jac(y(x))

(∂vβJ∂uαI

qJjβ∂xi

∂yj

). (4.30)

Proof. First of all, recall that Jkπ† = J1(πk−1)† is canonically isomorphic to the spaceof semi-basic form ω ∈ Λm

2 Jkπ (Proposition 3.28). We only have to write an arbitrary

semi-basic form ω ∈ Λm2 J

kπ in the two dierent systems of coordinates and use thetransformation rules (3.31) to get:

ω = p dmx+ pIiα duαI ∧ dm−1xi

= q dmy + qJjβ dvβJ ∧ dm−1yj

= Jac(y(x))

[q dmx+ qJjβ

∂xi

∂yj

(∂vβJ∂xk

dxk +∂vβJ∂uαI

duαI

)∧ dm−1xi

]

= Jac(y(x))

[(q +

∂vβJ∂xi

qJjβ∂xi

∂yj

)dmx+

(∂vβJ∂uαI

qJjβ∂xi

∂yj

)duαI ∧ dm−1xi

].

If we now compare the coecients of the rst and last expressions, we obtain the desiredresult.

While the kth jet bundle Jkπ projects over the lower order jet bundles (Diagram 4.1),the kth dual jet bundle is embedded into the (k+ 1)th dual jet bundle by means of thepullback of the ane projection πk+1,k (Diagram 4.4).

Proposition 4.30. The kth dual jet bundle of π, Jkπ†, together with the kth dual pro-jection, π†k,k−1, is a vector bundle over Jk−1π. Moreover, the induced coordinate systems(xi, uαI , p, p

Iiα ) are adapted to the vector bundle structure.


Jk+1π†

$$JJJJJJJJJ Jkπ†

""EEEEEEEEE? _

π∗k+1,koo · · ·

""EEEEEEEEE? _

π∗k,k−1oo J2π†

##GGGGGGGGG? _

π∗3,2oo J1π†

!!DDDDDDDD? _

π∗2,1oo

Jk+1ππk+1,k // Jkπ

πk,k−1 // · · · π3,2 // J2ππ2,1 // J1π

π1,0 // Eπ //M

Figure 4.4: Chain of dual jets

Denition 4.31. The reduced kth dual jet bundle of π is Jkπ := J1(πk−1) (see Denition3.26), which is isomorphic to Λm

2 Jk−1π/Λm

1 Jk−1π (Corollary 3.29).

Proposition 4.32. We have that:

1. Jkπ may be endowed with a structure of smooth manifold;

2. (Jkπ†, µ, Jkπ) is a smooth vector bundle of rank 1;

3. adapted coordinates (xi, uα) on E induce coordinates (xi, uαI , pIiα ) on Jkπ such that

µ(xi, uαI , p, pIiα ) = (xi, uαI , p

Iiα ), where (xi, uαI , p, p

Iiα ) are the induced coordinates on

Jkπ†.

Before we end this section, we summarize the important geometrical ingredients thathe dual jet bundle Jkπ† posses. In rst place, it has a canonical multisymplectic structurewhich is carried form the realization of Jkπ† as a semi-basic forms Λm

2 Jk−1π over Jk−1π

(see equations (3.32) and (3.33) and Denition 4.27). Moreover, its elements are naturallypaired with those of Jkπ (see Proposition 4.28). We recall that a form Ω is multi-symplectic if it is closed and if its contraction with a single tangent vector is injective,that is, iV Ω = 0 if and only if V = 0.

Denition 4.33. The Liouville or tautological m-form on Jkπ† is the form given by

Θω(V1, . . . , Vm) = ((π†k,k−1)∗ω)(V1, . . . , Vm), ω ∈ Jkπ†, V1, . . . , Vm ∈ TωJkπ†, (4.31)

where π†k,k−1 is the natural projection from Jkπ† to Jk−1π. The Liouville or canonicalmulti-symplectic (m+ 1)-form on Jkπ† is

Ω = − dΘ. (4.32)

Denition 4.34. The natural pairing between Jkπ and its dual Jkπ† is the bered mapΦ : Jkπ ×Jk−1π J

kπ† → ΛmM given by

Φ(jkxφ, ω) = (jk−1φ)∗jk−1x φ

ω. (4.33)

Let (xi, uαI , uαK) and (xi, uαI , p, p

Iiα ) denote adapted coordinates on Jkπ and Jkπ† re-

spectively, where |I| = 0, . . . , k − 1 and |K| = k. Then, the tautological form and thecanonical one are locally written

Θ = p dmx+ pIiα duαI ∧ dm−1xi and Ω = − dp ∧ dmx− dpIiα ∧ duαI ∧ dm−1xi, (4.34)

and the bered pairing between the elements of Jkπ and Jkπ† is locally written

Φ(xi, uαI , uαK , p, p

Iiα ) = (p+ pIiα u

αI+1i

) dmx. (4.35)


4.2 Higher Order Classical Field Theory

4.2.1 Variational Calculus

The dynamics in classical eld theory is specied giving a Lagrangian density: A La-grangian density is a mapping L : Jkπ → ΛmM . Fixed a volume form η on M , there isa smooth function L : Jkπ → R such that L = Lη.

Denition 4.35. Given a Lagrangian density L : Jkπ −→ ΛmM , the associated integralaction is the map AL : Γπ ×K −→ R given by

AL(φ,R) =

∫R

(jkφ)∗L, (4.36)


Denition 4.36. Let φ be a section of π. A (vertical) variation of φ is a curve ε ∈ I 7→φε ∈ Γπ (for some interval I ⊂ R) such that φε = ϕε φ ϕ−1

ε , where ϕε is the ow of a(vertical) π-projectable vector eld ξ on E.

When ξ is vertical, then its ow ϕε is an automorphism of ber bundles over theidentity for each ε ∈ I.

Denition 4.37. We say that φ ∈ Γπ is a critical or stationary point of the Lagrangianaction AL if and only if

d

dε[AL(φε, R)]

∣∣∣ε=0

=d

dε

[∫R

(jkφε)∗L] ∣∣∣∣

ε=0

= 0, (4.37)

for any vertical variation φε of φ whose associated vertical eld vanishes outside of π−1(R).

Lemma 4.38. Let φε = ϕε φ ϕ−1ε be a variation of a section φ ∈ Γπ. If ξ denotes the

innitesimal generator of ϕε, then

d

dε

[(jk(ϕε φ)∗xω

] ∣∣∣ε=0

= (jkφ)∗x(Lξ(k)ω), (4.38)

for any dierential form ω ∈ Ω(Jkπ).

Proof. From Proposition 4.18, we have that ξ(k) is the innitesimal generator of jkϕε. Wethen obtain by a direct computation,

(jkφ)∗x(Lξ(k)ω) = (jkφ)∗x

(d

dε

[(jkϕε)

∗ω] ∣∣∣ε=0

)=

d

dε

[(jkϕε jkφ)∗xω

] ∣∣∣ε=0

.

The following lemma will show to be useful in the variational derivation of the higher-order Euler-Lagrange equations.

4.2. HIGHER ORDER CLASSICAL FIELD THEORY 79

Lemma 4.39 (Higher-order integration by parts). Let R ⊂ M be a smooth compactregion and let f, g : R −→ R be two smooth functions. Given any multi-index J ∈ Nm,we have that∫

R

∂|J |f

∂xJg dmx = (−1)|J |

∫R

f∂|J |g

∂xJdmx+

∑If+Ig+1i=J

λ(If , Ig, J)

∫∂R

∂|If |f

∂xIf∂|Ig |g

∂xIgdm−1xi,

(4.39)where λ is given by the expression

λ(If , Ig, J) := (−1)|Ig | · |If |! · |Ig|!|J |!

· J !

If ! · Ig!. (4.40)

Proof. In this proof, we will use the shorthand notation fJ = ∂|J|f∂xJ

.We proceed by induction on the length l of the multi-index J . The case l = |J | = 0

is a trivial identity and the case l = |J | = 1 is the well known formula of integration byparts ∫

R

fig dmx =

∫∂R

fg dm−1xi −∫R

fgi dmx.

Thus, let us suppose that the result is true for any multi-index J ∈ Nm up to lengthl > 1, in order to show that it is also true for any multi-index K ∈ Nm of length l + 1.Let J and 1 ≤ j ≤ m such that J + 1j = K. We then have,∫

R

fJ+1jg dmx = −∫R

fJgj dmx+

∫∂R

fJg dm−1xj

= (−1)l+1

∫R

fgJ+1j dmx+

∫∂R

fJg dm−1xj

−∑

If+Igj+1i=J

λ(If , Igj , i)

∫∂R

fIfgIgj+1j dm−1xi,

where we have used the rst-order integration formula in rst place, to then apply theinduction hypothesis. We now multiply each member by (J(j) + 1)/(l+ 1) and sum overJ + 1j = K. Using the multi-index identity (A.7), we have∫

R

fKg dmx = (−1)l+1

∫R

fgK dmx+∑

J+1j=K

J(j) + 1

l + 1

∫∂R

fJg dm−1xj

−∑

J+1j=K

J(j) + 1

l + 1

∑If+Igj+1i=J

λ(If , Igj , i)

∫∂R

fIfgIgj+1j dm−1xi.

It only remain to rearrange properly the last two terms to express them in the statedform. Clearly,

∑J+1j=K

J(j) + 1

l + 1

∫∂R

fJg dm−1xj =

=∑

If+Ig+1i=K|Ig |=0

(−1)|Ig | · |If |! · |Ig|!|K|!

· K!

If ! · Ig!

∫∂R

∂|If |f

∂xIf∂|Ig |g

∂xIgdm−1xi.


The last term is a little bit more tricky,

∑J+1j=K

J(j) + 1

l + 1

∑If+Igj+1i=J

(−1)|Igj |+1 ·|If |! · |Igj |!|J |!

· J !

If ! · Igj !

∫∂R

fIfgIgj+1j dm−1xi =

=∑

If+Igj+1i+1j=K

(−1)|Igj |+1 ·|If |! · |Igj |!|K|!

· K!

If ! · Igj !

∫∂R

fIfgIgj+1j dm−1xi

=∑

If+Ig+1i=K|Ig |≥1

(−1)|Ig |∑

Igj+1j=Ig

Ig(j)

|Ig|· |If |! · |Ig|!|K|!

· K!

If ! · Ig!

∫∂R

fIfgIg dm−1xi

=∑

If+Ig+1i=K|Ig |≥1

(−1)|Ig | · |If |! · |Ig|!|K|!

· K!

If ! · Ig!

∫∂R

fIfgIg dm−1xi

where we have used the identity (A.7) again. The result is now clear.

In the following version of the higher-order Euler-Lagrange equations, we restrictourselves to vertical variations for simplicity, although it is possible to use also non-vertical variation like in Theorem 3.35.

Theorem 4.40 (The higher-order Euler-Lagrange equations). Given a ber section φ ∈Γπ, let us consider an innitesimal variation φε = ϕε φ of it such that the support Rof the associated vertical vector eld ξ is contained in a coordinate chart (xi). We thenhave that the variation of the Lagrangian action AL at φ is given by

d

dεAL(φε, R)

∣∣∣ε=0

=k∑|J |=0

(−1)|J |∫R

(j2kφ)∗(ξα

d|J |

dxJ∂L

∂uαJ

)dmx

+∑

Iξ+IL+1i=J

λ(Iξ, IL, J)

∫∂R

(j2kφ)∗(ξαIξ

d|IL|

dxIL∂L

∂uαJ

)dm−1xi

,(4.41)

where Rε = ϕε(R). Moreover, φ is a critical point of the Lagrangian action AL if andonly if it satises the higher-order Euler-Lagrange equations

(j2kφ)∗

k∑|J |=0

(−1)|J |d|J |

dxJ∂L

∂uαJ

= 0 (4.42)

on the interior of M , plus the boundary conditions

d|I|

dxI∂L

∂uαJ= 0, 0 ≤ |I| < |J | ≤ k, (4.43)

on the boundary ∂M of M .


Proof. Let us denote by ξ the vertical eld associated to the variation φε. By Proposition4.38, Cartan's formula L = d i+ i d and Proposition 4.18, we have that

d

dεAL(φε, R)

∣∣∣ε=0

=

∫R

(jkφ)∗(Lξ(k)L)

=

∫R

(jkφ)∗ d(iξ(k)L) +

∫R

(jkφ)∗iξ(k) dL

=

∫∂R

(jkφ)∗iξ(k)L+

∫R

(jkφ)∗(ξ(k)(L) dmx− dL ∧ iξ(k) dmx)

=

∫R

(jkφ)∗

k∑|J |=0

d|J |ξα

dxJ∂L

∂uαJ

dmx

If we now apply the higher-order integration by parts (4.39) and we take into accountthat Equation (4.13), we obtain that

d

dεAL(φε, R)

∣∣∣ε=0

=k∑|J |=0

(−1)|J |∫R

(j2kφ)∗(ξα

d|J |

dxJ∂L

∂uαJ

)dmx

+∑

Iξ+IL+1i=J

λ(Iξ, IL, J)

∫∂R

(j2kφ)∗(

d|Iξ|ξα

dxIξd|IL|

dxIL∂L

∂uαJ

)dm−1xi

,which is the rst statement of our theorem.

If we now suppose that R is contained in the interior of M , as ξ is null outside of R,so it is ξ(k) outside of R and, by smoothness, on its boundary ∂R. Thus, if φ is a criticalpoint of AL, we then must have that

d

dεAL(φε, R)

∣∣∣ε=0

=

∫R

(j2kφ)∗

ξα k∑|J |=0

(−1)|J |d|J |

dxJ∂L

∂uαJ

dmx = 0,

for any vertical eld ξ whose compact support is contained in π−1(R). We thus infer thatφ shall satisfy the higher-order Euler-Lagrange equations (4.42) on the interior of M .

Finally, if R has common boundary with M , we then have that

d

dεAL(φε, R)

∣∣∣ε=0

=k∑|J |=0

∑Iξ+IL+1i=J

λ(Iξ, IL, J)

∫∂R∩∂M

(j2kφ)∗(ξαIξ

d|IL|

dxIL∂L

∂uαJ

)dm−1xi = 0.

(4.44)As this is true for any vertical eld ξ whose compact support is contained in π−1(R), wededuce the boundary conditions (4.43).

4.2.2 Variational Calculus with Constraints

We consider a constraint submanifold i : C → Jkπ of codimension l, which is locallyannihilated by l functionally independent constraint functions Ψµ, where 1 ≤ µ ≤ l. Theconstraint submanifold C is supposed to ber over the whole ofM . Here one could use the


algorithm given in 4.22 so as to use the consequence up to order k given by the constraintsubmanifold C.

We now look for extremals of the Lagrangian action (4.36) restricted to those sectionsφ ∈ Γπ whose k-jet takes values in C (see [83, 121]). We will use the Lagrange multipliertheorem that follows.

Theorem 4.41 (Abraham, Marsden & Ratiu [2]). Let M be a smooth manifold, f :M −→ R be Cr, r ≥ 1, F a Banach space, g : M −→ F a smooth submersion andN = g−1(0). A point φ ∈ N is a critical point of f |N if and only if there exists λ ∈ F∗,called a Lagrange multiplier, such that φ is a critical point of f − 〈λ, g〉.

In order to apply the Lagrange multiplier theorem, we need to dene constraints asthe 0-level set of some function g. We congure therefore the following setting: choosethe smooth manifold M to be the space of local sections ΓRπ = φ : R ⊂ M → E :π φ = IdM, for some compact region R ⊂M . The Banach space F is the set of smoothfunctions C∞(R,Rl), provided with the L2-norm. The constraint function Ψ induces aconstraint function on the space of local sections ΓRπ by mapping each section φ to theevaluation of its k-lift by the constraint, that is,

g : φ ∈ ΓRπ 7→ Ψ jkφ ∈ C∞(R,Rl).

Note that the 0-level set N = g−1(0) is the set of sections whose k-lift takes values in theconstraint manifold C (over R).

We therefore obtain that a section φ : M −→ E is a regular critical point of the integralaction AL restricted to C if and only if there exists a Lagrange multiplier λ ∈ (C∞(R,Rl))∗

such that φ is a critical point of AL − 〈λ, g〉. A priori, we cannot assure that the pairing〈λ, g(φ)〉 has an integral expression of the type

∫RλµΨµ jkφ dmx for some functions

λµ : R −→ R. Henceforth, we shall suppose that it is the case.

Remark 4.42. In Theorem 4.41 appears some regularity conditions that exclude the so-called abnormal solutions. In general, given a critical point φ ∈ N = g−1(0) of f|N, the classical Lagrange multiplier theorem claims that there exists a nonzero element(λ0, λ) ∈ R×F∗ such that φ is a critical point of

λ0f − 〈λ, g〉 . (4.45)

Under the submersivity condition on g, that is φ is a regular critical point, it is possibleto guarantee that λ0 6= 0 and dividing by λ0 in (4.45) we obtain the characterizationof critical points given in Theorem 4.41. The critical points φ with vanishing Lagrangemultiplier, that is, λ0 = 0 are called abnormal critical points.

In the sequel we will only study the regular critical points, but our developmentsare easily adapted for the case of abnormality (adding the Lagrange multiplier λ0 andstudying separately both cases, λ0 = 0 and λ0 = 1).

Proposition 4.43 (Constrained higher-order Euler-Lagrange equations). Let φ ∈ Γπ bea critical point of the Lagrangian action AL given in (4.36) restricted to those sectionsof π whose kth lift take values in the constraint submanifold C ⊂ Jkπ. If the associatedLagrange multiplier λ is regular enough, then there must exist l smooth functions λµ :


R ⊂ M → R that satisfy together with φ the constrained higher-order Euler-Lagrangeequations

(j2kφ)∗

k∑|J |=0

(−1)|J |d|J |

dxJ

(∂L

∂uαJ− λµ

∂Ψµ

∂uαJ

) = 0. (4.46)

Proof. The proof is a direct application of Theorem 4.40 and Theorem 4.41.

4.2.3 The Skinner-Rusk formalism

The generalization of the Skinner-Rusk formalism to higher order classical eld theorieswill take place in the bered product

W0 = Jkπ ×Jk−1π Λm2 (Jk−1π). (4.47)

The results of this section constitute the main developments of our paper [27]. The rstorder case is covered in [50, 70]; see also [143, 144] for the original treatment by Skinnerand Rusk. The projection on the i-th factor will be denoted pr i (with i = 1, 2) and theprojection as ber bundle over Jk−1π will be πW0,Jk−1π = πk,k−1pr 1 (see Diagram 4.5). OnW0, adapted coordinate systems are of the form (xi, uαI , u

αK , p

I,iα , p), where |I| = 0, . . . , k−1

and |K| = k.

W0

pr1

wwoooooooooooooo

πW0,J

k−1π

pr2

))RRRRRRRRRRRRRRR

Jkππk,k−1

''OOOOOOOOOOOOO Λm2 (Jk−1π)

uullllllllllllll

Jk−1π

πk−1

M

Figure 4.5: The Skinner-Rusk framework

Assume that L : Jkπ −→ R is a Lagrangian function. Together with the pairing Φ(Proposition 4.28), we use this Lagrangian L to dene a dynamical function H (corre-sponding to the Hamiltonian) on W0:

H = Φ− L pr 1 . (4.48)

Consider the canonical multisymplectic (m + 1)-form Ω on Λm2 (Jk−1π) (Equation

(4.32)), whose pullback to W0 shall be denoted also by Ω. We dene on W0 the (m+ 1)-form

ΩH = Ω + dH ∧ η. (4.49)

In adapted coordinates

H = pI,iα uαI+1i

+ p− L(xi, uαI , uαK) (4.50)

ΩH = − dpI,iα ∧ duαI ∧ dm−1xi +

(pI,iα duαI+1i

+ uαI+1idpI,iα −

∂L

∂uαJduαJ

)∧ dmx,(4.51)

where |I| = 0, . . . , k − 1 and |J | = 0, . . . , k.


The dynamical equation

We search for an Ehresmann connection Γ in the ber bundle πW0,M : W0 −→ M whosehorizontal projector be a solution of the dynamical equation (see Section 1.1):

ihΩH = (m− 1)ΩH . (4.52)

We will show that such a solution does not exist on the whole W0. Thus, we need torestrict to the space on where such a solution exists, that is on

W1 = w ∈ W0 / ∃hw : TwW0 −→ TwW0 linear such that h2w = hw,

kerhw = (V πW0,M)w, ihwΩH(w) = (m− 1)ΩH(w). (4.53)

Remark 4.44. Equation (4.52) is a generalization of equations that usually appear in rstorder eld theories. In this particular case, from a given Lagrangian function L : J1π → Rwe may construct a unique (m+ 1)-form ΩL (the Poincaré-Cartan (m+ 1)-form). Hence,we have a geometrical characterization of the Euler-Lagrange equations for L as follows.Let Γ be an Ehresmann connection in π1,0 : J1π → M , with horizontal projector h.Consider the equation

ihΩL = (n− 1)ΩL. (4.54)

If h has locally the from

h

(∂

∂xi

)=

∂

∂xi+ Aαi

∂

∂uα+ Aαji

∂

∂uαj,

then a direct computation shows that equation (4.54) holds if and only if

(Aαi − uαi )

(∂2L

∂uαi ∂uβj

)= 0, (4.55)

∂L

∂uα− ∂2L


∂2L


∂2L

∂uβj ∂uαi

+ (Aβj − uβj )

∂2L

∂uα∂uβj= 0. (4.56)

(see [54]). If the lagrangian L is regular, then Eq. (4.55) implies that Aαi = uαi andtherefore Eq. (4.56) becomes

∂L

∂uα− ∂2L


∂2L


∂2L

∂uβj ∂uαi

= 0. (4.57)

Now, if σ(xi) = (xi, σα(x), σαi (x)) is an integral section of h we would have

uαi =∂σα

∂xiand Aαij =

∂σαi∂xj

,

which proves that Eq. (4.57) is nothing but the Euler-Lagrange equations for L.We may think Equation (4.52) as a generalization of equation 4.54 giving the Euler-

Lagrange equations for higher-order eld theories in a univocal way, as we will see.

In a local chart (xi, uαJ , pI,iα , p) ofW0, a horizontal projector hmust have the expression:

h =

(∂

∂xj+ AαJj

∂

∂uαJ+BIi

αj

∂

∂pI,iα+ Cj

∂

∂p

)⊗ dxj, (4.58)


where |I| = 0, . . . , k − 1 and |J | = 0, . . . , k. We then obtain that

ihΩH − (m−1)ΩH =

(BIiαi du

αI − AαIi dpI,iα + pI,iα duαI+1i

+ uαI+1idpI,iα −

∂L

∂uαJduαJ

)∧ dmx

=

(B iαi −

∂L

∂uα

)duα +

k−1∑|I′|=1

(BI′iαi −

∂L

∂uαI′

)duαI′ +

k−2∑|I|=0

pI,iα duαI+1i

−∑|K|=k

∂L

∂uαKduαK +

∑|I|=k−1

pI,iα duαI+1i

+k−1∑|I|=0

(uαI+1i− AαIi) dpI,iα

∧ dmx.

Equating this to zero and using Lemma A.5, we have that

AαIi = uαI+1i, |I| = 0, . . . , k − 1, i = 1, . . . ,m; (4.59)

B jαj =

∂L

∂uα; (4.60)

pI,iα =I(i) + 1

|I|+ 1

(∂L

∂uαI+1i

−BI+1ijαj +QIi

α

), |I| = 0, . . . , k − 2, i = 1, . . . ,m; (4.61)

pI,iα =I(i) + 1

|I|+ 1

(∂L

∂uαI+1i

+QIiα

), |I| = k − 1, i = 1, . . . ,m; (4.62)

where the Q's are arbitrary functions such that∑I+1i=J

I(i) + 1

|I|+ 1QIiα = 0, with |J | = 1, . . . , k. (4.63)

Remark 4.45. The ambiguity in the denition of the Legendre transform, and thereforeof the Cartan form, becomes apparent in the equations (4.61) and (4.62), as noted byCrampin and Saunders (see [140]). There are too many momentum variables to be relatedunivocally with the velocity counterpart. To x this, a choice of arbitrary functions Qsatisfying (4.63) must be done. The choice may be encoded as an additional geometricstructure, like a connection.

Applying (4.63) to (4.61) and (4.62), and using the identity (A.7), we nally obtainthe equations

AαIi = uαI+1i, with |I| = 0, . . . , k − 1, i = 1, . . . ,m; (4.64)

0 =∂L

∂uα−B j

αj; (4.65)∑I+1i=J

pI,iα =∂L

∂uαJ−BJj

αj , with |J | = 1, . . . , k − 1; (4.66)

∑I+1i=K

pI,iα =∂L

∂uαK, with |K| = k. (4.67)

Notice that equation (4.67) is the constraint that denes the space W1; and that(4.64), (4.65) an d(4.66) are conditions on coecients of the horizontal projectors h.


Note also that, for the time being, the A's with greatest order index and the C's remainundetermined, as well as the most part of theB's. From the denition ofW1, we know thatfor each point w ∈ W1 there exists a horizontal projector hw : TwW0 −→ TwW0 satisfyingequation (4.52). However, we cannot ensure that such hw, for each w ∈ W1, will takevalues in TwW1. Therefore, we impose the natural regularizing condition hw(TwW0) ⊂TwW1, ∀w ∈ W1. This latter condition is equivalent to having

h

(∂

∂xj

)( ∑I+1i=K

pI,iα −∂L

∂uαK

)= 0,

which in turn is equivalent (using (4.58) and (4.64)) to

∑I+1i=K

BIiαj =

∂2L

∂xj∂uαK+

k−1∑|I|=0

uβI+1j

∂2L

∂uβI ∂uαK

+∑|J |=k

AβJj∂2L

∂uβJ∂uαK

, (4.68)

with |K| = k. Thus, if the matrix of second order partial derivatives of L with respect tothe velocities of highest order (

∂2L

∂uβJ∂uαK

)(4.69)

is non-degenerate, then the highest order A's are completely determined in terms of thehighest order B's. In the sequel, we will say that the Lagrangian L : J1π −→ R is regularif, for any system of adapted coordinates the matrix, (4.69) is non-degenerate.

Up to now, no meaning has been assigned to the coordinate p. Consider the submani-foldW2 ofW1 dened by the restrictionH = 0. In other words,W2 is locally characterizedby the equation

p = L− pI,iα uαI+1i.

As before, we cannot ensure that a solution h of the dynamical equation (4.52) takesvalues in TW2. We thus impose to h the regularizing condition hw(TwW0) ⊂ TwW2,∀w ∈ W2, or equivalently h(∂/∂xj)(H) = 0. Therefore, the coecients of the linearmapping h are governed by the equations (4.64), (4.65), (4.66), (4.68) and in addition

Cj =∂L

∂xj+ AαJj

∂L

∂uαJ− AαI+1ij

pI,iα −BIiαju

αI+1i

. (4.70)

Note that, thanks to the Lemma A.5 and Equation (4.67), the terms with A's with multi-index of length k cancel out, and the A's with lower multi-index are already determined.So, in some sense, the C's depend only on the B's.

Description of the solutions

The relations (4.66) (with |J | = k − 1) and (4.68) can be seen as a system of linearequations with respect to the B's. When k = 1, equation (4.65) should be consideredinstead of equation (4.66). In the following, we are going to suppose that n = 1, since thedimension of the bres is irrelevant for our purposes and we may ignore it. The numberof B's with order k − 1 (with multi-index length k − 1) is given by(

m− 1 + k − 1

m− 1

)·m2


and the number of equations with such B's is(m− 1 + k

m− 1

)·m+

(m− 1 + k − 1

m− 1

).

An easy computation shows that the system is overdetermined if and only if k = 1 orm = 1 (examples 4.51 and 4.52), and completely determined when k = m = 2. In allother cases the system is underdetermined, but it still has maximal rank.

Proposition 4.46. Suppose that k ≥ 2 and m ≥ 2. Then, the system of linear equationswith respect to the B's

m∑j=1

BJjj =

∂L

∂uJ−∑

I+1i=J

pI,i; (4.71)

∑I+1i=K

BIij =

∂2L

∂xj∂uK+

k−1∑|I|=0

uI+1j

∂2L

∂uI∂uK+∑|J |=k

AJj∂2L

∂uJ∂uK; (4.72)

where |J | = k − 1, j = 1, . . . ,m and |K| = k, has maximal rank.

Proof. In a rst step, we are going to describe how to write the matrix of coecients.Then, we will select the proper columns of this matrix to obtain a new square matrix ofmaximal size. We nally shall prove that this matrix has maximal rank.

The matrix of coecients will be a rectangular matrix formed by 1's and 0's. Thecolumns will be indexed by the indexes of the B's, and the rows by the indexes of therst partial derivatives that appear in the equations (4.71) and (4.72). As BIi

j has threeindexes, the columns of the matrix of coecients will organized in a superior level by theindex i, in a middle level by the index j and in an inferior level by the multi-index I. Therows will be organized at the top by the index J for the rst equation, (4.71), and at thebottom by the index j and then by the multi-index K for the second equation, (4.72).

As the matrix of coecients has more columns than rows, we shall build a secondmatrix that has as many columns and rows as the matrix of coecients has rows. Todo that, we select a column of the matrix of coecients for each row index using thefollowing algorithm (for the sake of simplicity):

01 ForEach (j,K)

02 Define G=(I,i):I+1_i=K

03 If Cardinal(G)=1

04 Select the column (i,j,I)

05 ElseIf

06 Select a column (i,j,I) such that (I,i) is in G and i\neq j

07 EndIf

08 EndFor

09 ForEach J

10 If J(1)=k-1

11 Select the column (m,m,J)

12 ElseIf

13 Select the column (1,1,J)

14 EndIf

15 EndFor


Now, this matrix being dened and since it is full of 0's and has only few 1's, we aregoing to develop its determinant by rows and columns. Notice that the columns selectedat line 6 have only one 1 each, thus we can cross out the rows an columns related to these1's. Now the rows at the bottom part of the remaining matrix (related to the secondequations) have only one 1 each, thus we can also cross out the rows an columns relatedto these 1's. Now, the remaining matrix has the property of having only one 1 per columnand row (there must be at least one 1 per row and column, and no two 1's may be at thesame row or column), thus its determinant is not zero and the matrix of coecients hasmaximal rank.

Example 4.47. If we consider the simple case of third order (k = 3) with two independentvariables (m = 2), then we will obtain a system of 11 equations with 12 unknowns. Thematrix of coecients will take the form

1 0 0 0 0 0 0 0 0 1 0 00 1 0 0 0 0 0 0 0 0 1 00 0 1 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 0 0 00 1 0 1 0 0 0 0 0 0 0 00 0 1 0 1 0 0 0 0 0 0 00 0 0 0 0 1 0 0 0 0 0 00 0 0 0 0 0 1 0 0 0 0 00 0 0 0 0 0 0 1 0 1 0 00 0 0 0 0 0 0 0 1 0 1 00 0 0 0 0 0 0 0 0 0 0 1

where the columns are labeled in order by: (1, 1, 11 + 11), (1, 1, 11 + 12), (1, 1, 12 + 12),(1, 2, 11 + 11), (1, 2, 11 + 12), (1, 2, 12 + 12), (2, 1, 11 + 11), (2, 1, 11 + 12), (2, 1, 12 + 12),(2, 2, 11+11), (2, 2, 11+12), (2, 2, 12+12); and where the rows are ordered by: 11+11, 11+12,12 +12, (1, 11 +11 +11), (1, 11 +11 +12), (1, 11 +12 +12), (1, 12 +12 +12), (2, 11 +11 +11),(2, 11 + 11 + 12), (2, 11 + 12 + 12), (2, 12 + 12 + 12). The previous algorithm would selectall the columns but the eleventh (which corresponds to the label (2, 2, 11 + 12)) in thefollowing order: (1, 1, 11 + 11), (2, 1, 11 + 11), (2, 1, 11 + 12), (2, 1, 12 + 12), (1, 2, 11 + 11),(1, 2, 11 + 12), (1, 2, 12 + 12), (2, 2, 12 + 12), (2, 2, 11 + 11), (1, 1, 11 + 12), (1, 1, 12 + 12).Note that the resulting matrix is regular.

The problem get worst with a little increment of the order or the number of indepen-dent variables. For instance the case k = 5 and m = 6 gives a system of 1.638 equationsand 4.536 unknowns.

Another way to interpret the tangency condition (4.68) is the following one: Let ussuppose we are dealing with a rst order Lagrangian (example 4.51, equation (4.89)). Onecould apply the theory of connections to the Lagrangian setting and the Hamiltonian oneas separate frameworks. We know that they must be related by means of the Legendretransform and so are the horizontal projectors induced by these connections. Thus,equation (4.89) is nothing else than the relation between the coecients of these horizontalprojectors.


The reduced mixed space W2

In section 4.2.3 we reduced the space W1 to W2 by considering the constraint H = 0,which is a way of interpreting the coordinate p as the Hamiltonian function. But W2

is not a mere instrument to get rid o the coordinate p or the coecients Cj. As thepremultisymplectic form ΩH , it encodes the dynamics of the system and, when L isregular, it is a multisymplectic space. Indeed, when k = 1, W2 is dieomorphic to J1π(cf. de León et al. [50]), which is not true for higher order cases.

Proposition 4.48. Let W2 = w ∈ W1 : H(w) = 0 and dene the (m + 1)-form Ω2

as the pullback of ΩH to W2 by the natural inclusion i : W2 → W0, that is Ω2 = i∗(ΩH).Suppose that dimM > 1, then, the (m+ 1)-form Ω2 is multisymplectic if and only if L isregular.

Proof. First of all, let us make some considerations. By denition, Ω2 is multisymplecticwhenever Ω2 has trivial kernel, that is,

if v ∈ TW2, ivΩ2 = 0 ⇐⇒ v = 0 .


if v ∈ i∗(TW2), ivΩH |i∗(TW2) = 0 ⇐⇒ v = 0 .

Let v ∈ TW0 be a tangent vector whose coecients in an adapted basis are given by

v = λi∂

∂xi+ AαJ

∂

∂uαJ+BIi

α

∂

∂pIiα+ C

∂

∂p.

Using the expression (4.51), we may compute the contraction of ΩH by v,

ivΩH = −BIiα duαI ∧ dm−1xi + AαI dpIiα ∧ dm−1xi − λj dpIiα ∧ duαI ∧ dm−2xij

+(AαI+1i

pIiα +BIiα u

αI+1i− AαJ ∂L

∂uαJ

)dmx

−λj(pIiα duαI+1i

+ uαI+1idpIiα − ∂L

∂uαJduαJ

)∧ dm−1xj.

(4.73)

On the other hand, if we now suppose that v is tangent toW2 inW0, that is v ∈ i∗(TW2),we then have that

d

( ∑I+1i=K

pIiα −∂L

∂uαK

)(v) = 0 and dH(v) = 0, (4.74)

which leads us to the following relations for the coecients of v,∑I+1i=K

BIiα = λi

∂2L

∂xi∂uαK+ AβJ

∂2L

∂uβJ∂uαK

and (4.75)

AαI+1ipIiα +BIi

α uαI+1i

+ C − λi ∂L∂xi− AαJ ∂L

∂uαJ= 0. (4.76)

It is important to note that thanks to Lemma A.2 and the equation (4.67) which denesW1 (and hence W2), the terms in (4.73) and (4.76) involving A's with multi-index oflength k cancel each other out.


These considerations being made, suppose that Ω2 is multisymplectic and, by reductioad absurdum, suppose in addition that L is not regular, which means that the matrix(

∂2L

∂uβK′∂uαK

)has non-trivial kernel. Let v ∈ TW0 be a tangent vector such that all its coecients arenull except the A's of highest order which are in such a way they are mapped to zero bythe hessian of L. Such a vector v fullls the restrictions (4.75) and (4.76), thus it mustbe tangent to W2 in W0, v ∈ i∗(TW2). But, as ivΩH has no A's of highest order, it mustbe zero, ivΩH = 0, which is a contradiction.

Conversely, let us suppose that L is regular, then equation (4.67) denes implicitlythe coordinates uαK as functions of the other coordinates. That is, locally there existfunctions fαK(xi, uαI , p

I,iα ) such that uαK = fαK on i(W2). Furthermore, given a system of

adapted coordinates (xi, uαI , uαK , p

I,iα , p) onW0, (xi, uαI , p

I,iα ) denes a coordinate system on

W0 and the inclusion is given by:

(xi, uαI , pI,iα ) ∈ W2 → (xi, uαI , f

αK , p

I,iα , L−

k−2∑|I|=0

pI,iα uαI+1i−∑|I|=k−1

pI,iα fαI+1i

) ∈ W0.

From equation (4.51), we can compute an explicit expression of the (m + 1)-form Ω2 inthis coordinate system,

Ω2 = −k−1∑|I|=0

dpI,iα ∧ duαI ∧ dm−1xi

+

k−2∑|I|=0

(pI,iα duαI+1i

+ uαI+1idpI,iα

)−

k−1∑|I|=0

∂L

∂uαIduαI

∧ dmx

+

∑|I|=k−1

(pI,iα dfαI+1i

+ fαI+1idpI,iα

)−∑|K|=k

∑I+1i=K

pI,iα dfαK

∧ dmx,

where we have used equation (4.67) in the last term. Note that, by Lemma A.2, the rstand last terms of the last bracket cancel each other out. Now,

i∂/∂xjΩ2 = dpI,iα ∧ duαI ∧ dm−2xij − [. . . ] ∧ dm−1xj

i∂/∂uαI Ω2 = dpI,iα ∧ dm−1xi +

∑J+1j=I

pJjα −∂L

∂uαI

dmx

i∂/∂pI,iα Ω2 = duαI ∧ dm−1xi + uαI+1idmx,

where 0 ≤ |I| ≤ k−1. We deduce from here that the kernel of Ω2 is trivial, ker Ω2 = 0,and Ω2 is multisymplectic.

Note 4.49. In the particular case when dimM = 1, the Lagrangian function L : Jkπ −→ Ris regular if and only if the pair (Ω2, τ

∗W2,M

dt) is a cosymplectic structure onW2. We recallthat a cosymplectic structure on a manifold N of odd dimension 2n + 1 is a pair whichconsists of a closed 2-form Ω and a closed 1-form η such that η ∧ Ωn is a volume form.


We remark that, if the Lagrangian L is regular or (from Proposition 4.46) if k,m > 1,then there locally exist solutions h of the dynamical equations (4.52) on W2 that giverise to connections Γ in the bration πW0M : W0 −→ M along the submanifold W2 (seeSection 1.1). In such a case, a global solution is obtained using partitions of the unity,and we obtain by restriction a connection Γ, with horizontal projector h, in the brebundle πW2M : W2 −→ M , which is a solution of equation (4.52) when it is restricted toW2 (in fact, we have a family of such solutions).

In some cases, but only when dimM = 1 or k = 1, it would be necessary to considera subset W3 dened in order to satisfy the tangency conditions (4.68) and (4.70):

W3 = w ∈ W2 / ∃hw : TwW0 −→ TwW2 linear such that h2w = hw,

kerhw = (V πW0,M)w, ihwΩH(w) = (m− 1)ΩH(w).

We will assume that W3 is a submanifold of W2. If hw(TwW0) is not contained in TwW3,we go to the third step, and so on. At the end, and if the system has solutions, wewill nd a nal constraint submanifold Wf , bered over M (or over some open subsetof M) and a connection Γf in this bration such that Γf is a solution of equation (4.52)restricted to Wf .

In any case, one obtains the Euler-Lagrange equations. In the following result, Wf

denotes the nal constraint manifold, which is W2 when k,m > 1, and h the horizontalprojector of a connection in πW2,M : Wf −→ M along Wf , which is solution of thedynamical equation.

Proposition 4.50. Let σ be a section of πWf ,M : Wf −→ M and denote σ = i σ,where i : Wf → W0 is the canonical inclusion. If σ is an integral section of h, then σ isholonomic, in the sense that

pr 1 σ = jk(πWf ,E σ), (4.77)

and satises the higher-order Euler-Lagrange equations:

j2k(πWf ,E σ)∗

k∑|J |=0

(−1)|J |d|J |

dxJ∂L

∂uαJ

= 0. (4.78)

Proof. If σ = (xi, σαJ , σI,iα , σ) is an integral section of h, then

∂σαJ∂xj

= AαJj,∂σIiβ∂xj

= BIiβj and

∂σ

∂xj= Cj,

where the A's, B's and C's are the coecients given in (4.58). From equation (4.64), wehave that σ is holonomic, in the sense that σαI+1i

= ∂σαI /∂xi. On the other hand, using

the equations (4.65), (4.66) and (4.67), we obtain the relations (where φ = pr1 σ):

0 =∂L

∂uα φ− ∂σ j

α

∂xj; (4.79)∑

I+1i=J

σI,iα =∂L

∂uαJ φ− ∂σJjα

∂xj, with |J | = 1, . . . , k − 1; (4.80)

∑I+1i=K

σI,iα =∂L

∂uαK φ, with |K| = k. (4.81)


From the equations (4.79) and (4.80) for |J | = 1 we get

0 =∂L

∂uα φ− ∂σ j

α

∂xj

= (j0φ)∗∂L

∂uα−∑|I|=1

(j1φ)∗(

d|I|

dxI∂L

∂uαI

)+∑|I|=1

∑i

∂|I|

∂xI∂σIiα∂xi

.

Applying now Lemma A.2 on the last term and repeating this process until |I| = k − 1we reach

0 = (j0φ)∗∂L

∂uα−∑|I|=1

(j1φ)∗(

d|I|

dxI∂L

∂uαI

)+∑|J |=2

∂|J |

∂xJ

∑I+1i=J

σIiα

= (j0φ)∗∂L

∂uα−∑|I|=1

(j1φ)∗(

d|I|

dxI∂L

∂uαI

)+∑|I|=2

(j2φ)∗(

d|I|

dxI∂L

∂uαI

)−∑|I|=2

∑i

∂|I|

∂xI∂σIiα∂xi

=k−1∑|I|=0

(−1)|I|(j|I|φ)∗(

d|I|

dxI∂L

∂uαI

)− (−1)k−1

∑|I|=k−1

∑i

∂|I|

∂xI∂σIiα∂xi

=k−1∑|I|=0

(−1)|I|(j|I|φ)∗(

d|I|

dxI∂L

∂uαI

)− (−1)k−1

∑|K|=k

∂|K|

∂xK

∑I+1i=K

σIiα ,

where by abuse of notation jlφ = jk+l(πWf ,E σ). Finally, it only rest to use equation(4.81) to prove the desired result.

Examples

First, we are going to study the particular cases when k = 1 andm = 1, which correspondto the First Order Classical Field Theory and to the Higher Order Mechanical Systems,respectively. Theoretic results for these cases are very well known [15, 50, 70, 103] andwe are only going to recover these results from our general setting. In addition, theseparticular cases will clarify the general procedure.

Example 4.51 (First order Lagrangians (k = 1)). Let us suppose that k = 1, which corre-sponds to the case of rst order Lagrangians. In that case the velocity-momentum space isW0 = J1π ⊗E Λm

2 E, with adapted coordinates (xi, uα, uαi , p, piα). The premultisymplectic

(m+ 1)-form would be

ΩH = − dpiα ∧ duα ∧ dm−1xi +

(piα duαi + uαi dpiα −

∂L

∂uαduα − ∂L

∂uαiduαi

)∧ dmx, (4.82)

and horizontal projectors on TW0 would have locally the form:

h =

(∂

∂xj+ Aαj

∂

∂uα+ Aαij

∂

∂uαi+B i

αj

∂

∂piα+ Cj

∂

∂p

)⊗ dxj. (4.83)

Solutions of the dynamical equation would satisfy the relationsm∑j=1

B jαj =

∂L

∂uα; (4.84)

piα =∂L

∂uαi, for i = 1, . . . ,m; (4.85)

Aαi = uαi , for i = 1, . . . ,m; (4.86)


from which we deduce the Euler-Lagrange equations

j2(πW2,M σ)∗

(∂L

∂uα−

m∑i=1

d

dxi∂L

∂uαi

)= 0, (4.87)

where W2 is dened by

W2 =

(xi, uα, uαi , p, p

iα) ∈ W1 : piα =

∂L

∂uαi, p = L−

m∑i=1

piui

. (4.88)

We then obtain the tangency conditions:

B iαj =

∂2L

∂xj∂uαi+ uβj

∂2L

∂uβ∂uαi+

m∑l=1

Aβlj∂2L

∂uβl ∂uαi

, (4.89)

Cj =∂L

∂xj+ uαj

∂L

∂uα−B i

αjuαi . (4.90)

Note that (4.89) is the relation that would appear between the coecients of a Lagrangianand a Hamiltonian setting through the Legendre transform. For simplicity, suppose thatn = 1 and ignore the α's and β's that appear above. Consider the linear system ofequations with respect to the B's formed by equations (4.84) and (4.89). This system isoverdetermined since it has m2 + 1 equations and only m2 variables (Bi

j).

Example 4.52 (Higher order mechanical systems (m = 1)). Let us suppose that m = 1,which corresponds to the case of mechanical systems. In that case the velocity-momentumspace is W0 = Jkπ ×Jk−1π Λm

2 (Jk−1π). Since here a multi-index J is of the form (l) with1 ≤ l ≤ k, we change the usual notation for coordinates to

uαJ −→ uα|J | and pI,1α −→ p|I|+1α ,

and we adapt the remaining objects to this notation. So adapted coordinates on W0 areof the form (x, uα, uαl , p, p

lα), where l = 1, . . . , k. The premultisymplectic (m + 1)-form

would be

ΩH = −k−1∑l=0

dpl+1α ∧ duαl +

k∑l=1

(plα duαl + uαl dplα

)∧ dx−

k∑l=0

∂L

∂uαlduαl ∧ dx, (4.91)

and horizontal projectors on TW0 would have locally the form:

h =

(∂

∂x+

k∑l=0

Aαl∂

∂uαl+

k∑l=1

Blα

∂

∂plα+ C

∂

∂p

)⊗ dx. (4.92)

Solutions of the dynamical equation would satisfy the relations

B1α =

∂L

∂uα; (4.93)

plα =∂L

∂uαl−Bl+1

α , for l = 1, . . . , k − 1; (4.94)

pkα =∂L

∂uαk; (4.95)

Aαl = uαl+1, for l = 0, . . . , k − 1. (4.96)


which we use to get the Euler-Lagrange equations

j2k(πW2,M σ)∗

(k∑l=0

(−1)ldl

dxl∂L

∂uαl

)= 0, (4.97)

where W2 is dened by

W2 =

(xi, uα, uαl , p, p

lα) ∈ W1 : pkα =

∂L

∂uαk, p = L−

k∑l=1

plαuαl

. (4.98)

We then obtain the tangency conditions:

Bkα =

∂2L

∂x∂uαk+

k−1∑l=0

uβl+1

∂2L

∂uβl ∂uαk

+ Aβk′∂2L

∂uβk′∂uαk

= 0; (4.99)

C =∂L

∂x+

k−1∑l=0

uαl+1

∂L

∂uαl+ Aαk

∂L

∂uαk−

k∑l=1

(Aαl p

lα +Bl

αjuαl

). (4.100)

Note that, thanks to equation (4.95), the terms in (4.100) with coecient Ak cancel out.Now, for simplicity, suppose that n = 1 and ignore the α's and β's that appear above.Consider the linear system of equations with respect to the B's formed by equations(4.94) (with l = k−1) and (4.99). This system is overdetermined since it has 2 equationsand only one variable (Bk).

Example 4.53 (The loaded and clamped plate). Let us setM = R2 and E = R2×R = R3,and consider the Lagrangian

L(x, y, u, ux, uy, uxx, uxy, uyy) =1

2(u2

xx + 2u2xy + u2

yy − 2qu),

where q = q(x, y) is the normal load on the plate. Given a regular region R of the plane,we look for the extremizers of the functional I(u) =

∫RL such that u = ∂u/∂n = 0

on the border ∂R, where n is the normal exterior vector. The Euler-Lagange equationassociated to the problem is

uxxxx + 2uxxyy + uyyyy = q. (4.101)

Written in the multi-index notation, the Lagrangian has the form

L(j2φ) =1

2(u2

(2,0) + 2u2(1,1) + u2

(0,2) − 2qu)

and the Euler-Lagrange equation reads

u(4,0) + 2u(2,2) + u(0,4) = q.

The velocity-momentum space is W0 = J2π ×J1π Λ22(J1π), with adapted coordinates

(x, y, ux, uy, uxx, uxy, uyy, p, px, py, pyy, pxy, pyx, pyy). It is straightforward to write down

the premultisymplectic 3-form and a general horizontal projector on TW0, so we are not


going to do it here. Even so, the coecients of solutions of the dynamical equation wouldsatisfy the relations

B ,xx +B ,y

y = −2q−px = Bx,x

x +Bx,yy

−py = By,xx +By,y

y

pxx = uxxpxy + pyx = 2uxy

pyy = uyy

(4.102)

where the latter ones are the equations that dene W1. The tangency condition on W1

gives us the relations

Bx,xx = Axx,x

Bx,yx +By,x

x = 2Axy,xBy,yx = Ayy,x

Bx,xy = Axx,y

Bx,yy +By,x

y = 2Axy,yBy,yy = Ayy,y

(4.103)

from where we can see that the Lagrangian is regular, since

(∂2L

∂uK∂uK′

)|K|=|K′|=2

=

1 0 00 2 00 0 1

. (4.104)

Finally, we remark that the middle equations of (4.102) and (4.103) form a 8 × 8 linearsystem of equations on the B's, which is completely determined.

Example 4.54 (The Camassa-Holm equation). In 1993, Camassa and Holm introducedthe following completely integrable bi-Hamiltonian equation (see [23]):

vt − vyyt = −3vvy + 2vyvyy + vvyyy, (4.105)

which is used to model the breaking waves in shallow waters as the Kortewegde Vriesequation. But, as the former is of higher order, we are going to use it as example.

The CH equation (4.105) is expressed in terms of the Eulerian or spatial velocity eldu(y, t), and it is the Euler-Poincaré equation of the reduced Lagrangian

l(v) =1

2

∫ (v2 + v2

y

)dy. (4.106)

To give a multisymplectic approach to the problem, as Kouranbaeva and Shkoller did (see[112]), we must express the CH equation (4.105) in Lagrangian terms. Thus, we shall usethe Lagrangian variable u(x, t) that arises as the solution of

∂u(x, t)

∂t= v(u(x, t), t). (4.107)

The independent variables (x, t) are coordinates for the base space M = S1×R, and thedependent variable u(x, t) is a ber coordinate for the total space E = S1 × R × R =S1 × R2. The Lagrangian action is now written as

L(x, t, u, ux, ut, uxx, uxt, utt) =1

2(uxu

2t + u−1

x u2xt) (4.108)


The coecients of a horizontal projector which is solution of the dynamical equation mustsatisfy

B ,xx +B ,t

t = 0px = 1/2(u2

t − (uxt/ux)2)− (Bx,x

x +Bx,tt )

pt = uxut − (Bt,xx +Bt,t

t )pxx = 0

pxt + ptx = uxt/uxptt = 0

(4.109)

where the last three are the equations that dene W1. The tangency condition on W1

gives us the relations

Bx,xx = 0

Bx,tx +Bt,x

x = −u−1x uxxuxt + Axt,xu

−1x

Bt,tx = 0

Bx,xt = 0

Bx,tt +Bt,x

t = −(uxt/ux)2 + Axt,tu

−1x

Bt,tt = 0

(4.110)

from where we can see that the Lagrangian is clearly singular, since

(∂2L

∂uK∂uK′

)|K|=|K′|=2

=

0 0 00 u−1

x 00 0 0

(4.111)

Again, we may form a completely determined system of linear equations on the B's withthe corresponding relations of (4.102) and the equations (4.110).

Example 4.55 (First order Lagrangian as second order). For the sake of simplicity, letsuppose that n = 1. Given a rst order Lagrangian L : J1π −→ R, extend it to asecond order Lagrangian, L = L π2,1. Consider the rst and second order velocity-momenta mixed spaces W 1

0 = J1π ×E Λm2 E and W 2

0 = J2π ×J1π Λm2 J

1π, with adaptedcoordinates (xi, u, ui, p, p

i) and (xi, u, ui, uK , p, pi, pij) (with |K| = 2), respectively. Let

π2,10 : W 2

0 −→ W 10 be the natural projection (Diagram 4.6).

W 20

π2,10 //

''OOOOOOOOOOOOO

W 10

&&NNNNNNNNNNNNN

J2π

π2,1 //

L

''OOOOOOOOOOOOOO J1π //

π1

''OOOOOOOOOOOOO

L

E

π

R M

Figure 4.6: The 1st and 2nd order Lagrangian settings

We are going to apply the theory we have developed here to the systems given byeach Lagrangian. Consider the premultisymplectic forms ΩH and ΩH , where H and Hare the corresponding dynamical functions (equations (4.48) and (4.49)). Let h and h


denote solutions of the respective dynamical equations on (W 10 ,ΩH) and (W 2

0 ,ΩH). Theywould locally have the form

h =

(∂

∂xj+ Aj

∂

∂u+ Aij

∂

∂ui+Bi

j

∂

∂pi+ Cj

∂

∂p

)⊗ dxj,

h =

(∂

∂xj+ Aj

∂

∂u+ Aij

∂

∂ui+ AKj

∂

∂uK+ Bi

j

∂

∂pi+ Bki

j

∂

∂pki+ Cj

∂

∂p

)⊗ dxj,

where |K| = 2. We then obtain the relations

Bjj =

∂L

∂u, (4.112)

pi =∂L

∂ui, (4.113)

Ai = ui, (4.114)

for (W 10 ,ΩH ,h); and

Bjj =

∂L

∂u, (4.115)

pi =∂L

∂ui− Bij

j , (4.116)

pij + pji = (1i + 1j)! ·∂L

∂u1i+1j

= 0, (4.117)

Ai = ui, (4.118)

Aij = u1i+1j , (4.119)

for (W 20 ,ΩH , h). Equations (4.113) and (4.117), together with H = 0 and H = 0, dene

the corresponding submanifolds W 12 and W 2

2 of W 10 and W 2

0 .We notice that, even though L is in some sense the same Lagrangian than L, a solution

of the dynamical equation on W 10 may be easily determined, while in W 2

0 the space ofsolutions has grown (there are more coecients to be determined). We thus infer fromhere, that a solution h of the dynamical equation in W 2

0 must satisfy an extra condition.Since p = L− piui + 0 in W 2

2 , the projection π2,10 maps W 2

2 to W 12 . We therefore impose

to a solution h of the dynamical equation along W 22 to be in addition projectable to a

solution h of the dynamical equation along W 12 . In such a case, we would have that

Bijj = 0 (4.120)

which implies that the following equation

pi =∂L

∂ui(4.121)

is now a restriction in W 22 . So, by tangency condition, we get

Bij =

∂2L

∂xj∂ui+ uj

∂2L

∂u∂ui+ u1k+1j

∂2L

∂uk∂ui+ 0 =

d

dxj∂L

∂ui. (4.122)


Combining this with equation (4.112), we nally obtain

∂L

∂u− d

dxj∂L

∂uj= 0, (4.123)

which is the Euler-Lagrange equation.It is worth to remark here that, at this time, the Euler-Lagrange equation has not

been deduced by the process shown in the proof of Proposition 4.50, but directly from theprojectability condition, although the previous Euler-Lagrange equation may be recoveredfrom any of the two settings.

4.2.4 Constraints within the Skinner-Rusk Formalism

As in the previous section, we begin by considering a constraint submanifold i : C → Jkπof codimension l, which is locally annihilated by l functionally independent constraintfunctions Ψµ, where 1 ≤ µ ≤ l. The constraint submanifold C is supposed to ber over thewhole ofM and it is not necessarily generated from a previous constraint submanifold bythe process shown in Remark 4.22. We dene in the restricted velocity-momentum spaceW0 = w ∈ W : H(w) = 0 the constrained velocity-momentum space W C

0 = pr−11 (C),

which is a submanifold of W0, whose induced embedding and whose constraint functionswill still be denoted i : W C

0 → W and Ψµ, where 1 ≤ µ ≤ l. The rst order case k = 1 istreated in [33].

The following proposition allows us to work in local coordinates on the unconstrainedvelocity-momentum space W , as it is done in [11].

Proposition 4.56. Given a point w ∈ W C0 , let X ∈ Λm

d (TwWC0 ) be a decomposable

multivector and denote its image, i∗(X) ∈ Λm(TwW ), by X. The following statementsare equivalent:

1. iXΩC0(Y ) = 0 for every Y ∈ TwW C0 ;

2. iXΩ ∈ T 0wW

C0 ;

where T 0wW

C0 is the annihilator of i∗(TwW C

0 ) in TwW .

We therefore look for solutions of the constrained dynamical equation

(−1)miXΩ = −λµ dΨµ − λ dH, (4.124)

where X is a tangent multivector eld along W C0 , the λ

µ's and λ are Lagrange multipliersto be determined. Here, the coecient (−1)m is used for technical purposes.

Remark 4.57. It should be said that the Lagrange multipliers that appear in the dynamicalequation (4.124) have a dierent nature that the ones that appear in Proposition 4.43.The former are locally dened on W , while the latter are locally dened onM . Althoughthey coincide on the integral sections σ ∈ ΓπW,M of a solutionX of the dynamical equation(4.124), since its Lagrangian part σ1 = pr 1 σ satises the constrained Euler-Lagrangeequation (4.46) with λµ = λµ σ (cf. Proposition 4.59).


Let X ∈ Λm(TwW ) be a decomposable m-vector at a given point w ∈ W , that is,X = X1 ∧ · · · ∧ Xm for m tangent vectors Xi ∈ TwW , which have the form

Xj =∂

∂xj+ AαJj

∂

∂uαJ+BIi

αj

∂

∂pIiα+ Cj

∂

∂p(4.125)

in a given adapted chart (xi, uαJ , pIiα , p). A straightforward computation gives us

(−1)miX( dpIiα ∧ duαI ∧ dm−1xi) = (AαIiBIjαj − AαIjB

Ijαi) dxi + AαIi dp

Iiα −BIi

αi duαI (4.126)

and(−1)miX( dp ∧ dmx) = dp− Ci dxi. (4.127)

Applying the above equations to the dynamical one (4.124), we obtain the relations

coecients in dp : 1 = λ;coecients in dpIiα : AαIi = λuαI+1i

;coecients in duαJ B i

αi = λ ∂L∂uα− λµ ∂Ψµ

∂uα;

BIiαi = λ

(∂L∂uαI−∑

J+1j=IpJjα

)− λµ ∂Ψµ

∂uαI;

0 = λ(

∂L∂uαK−∑

J+1j=KpJjα

)− λµ ∂Ψµ

∂uαK;

coecients in dxj : AαIiBIiαj − AαIjBIi

αi + Cj = λ ∂L∂xj− λµ ∂Ψµ

∂xj.

Thus, a decomposable m-vector X ∈ Λm(TwW ) at a point w ∈ W is a solution of thedynamical equation

(−1)miXΩ = −λµ dΨµ − dH, (4.128)

if for any adapted chart (xi, uαJ , pIiα , p), the coecients of X and the point w satisfy the

equations

AαIi =uαI+1i, with |I| = 0, . . . , k − 1, i = 1, . . . ; (4.129)

0 =∂L

∂uα− λµ

∂Ψµ

∂uα−B j

αj; (4.130)∑I+1i=J

pIiα =∂L

∂uαJ− λµ

∂Ψµ

∂uαJ−BJj

αj , with |J | = 1, . . . , k − 1; (4.131)

∑I+1i=K

pIiα =∂L

∂uαK− λµ

∂Ψµ

∂uαK, with |K| = k; (4.132)

Cj =∂L

∂xj− λµ

∂Ψµ

∂xj+

k−1∑|J |=0

uαJ+1j

(∂L

∂uαI− λµ

∂Ψµ

∂uαJ−∑

I+1i=J

pIiα

)− uαI+1i

BIiαj.

(4.133)

Because of the Lagrange multipliers λµ, we cannot describe the submanifold of W C0

where solutions X of the constrained dynamical equation (4.128) exist, like it has beendone in (4.67) for the unconstrained dynamical equation (4.52) . Therefore, we need toget rid o of them. Consider the more concise expression for the equations of dynamics(4.130), (4.131) and (4.132)∑

I+1i=J

pIiα =∂L

∂uαJ− λµ

∂Ψµ

∂uαJ−BJj

αj , with |J | = 0, . . . , k, (4.134)


where the rst summation term is understood to be void when |J | = 0, as well as itis the last one when |J | = k. We now suppose that the constraints Ψµ are of thetype uα

J= Φα

J(xi, uα

J), where uα

Jare some constrained coordinates which depend on the

free coordinates (xi, uαJ) through the functions Φα

J. Thus, the constraint have the form

ΨαJ(xi, uαJ) = uα

J−Φα

J(xi, uα

J) = 0. So, writing again the previous equation (4.134) for the

dierent sets of coordinates, the ones that are free and the ones that are not, we obtain∑I+1i=J

pIiα =∂L

∂uαJ

− λJα −BJjαj , with |J | = 0, . . . , k; (4.135)

∑I+1i=J

pIiα =∂L

∂uαJ

+ λJα∂Φα

J

∂uαJ

−BJjαj , with |J | = 0, . . . , k. (4.136)

Substituting −λJα from (4.135) into (4.136), we have that

∑I+1i=J

pIiα +

∑I+1i=J

pIiα

∂ΦαJ

∂uαJ

=∂L

∂uαJ

+∂L

∂uαJ

∂ΦαJ

∂uαJ

−BJjαj −B

Jjαj

∂ΦαJ

∂uαJ

, with |J | = 0, . . . , k.

(4.137)

Note that, when |J | = k, the term BJjαj disappears, but B

Jjαj

∂ΦαJ

∂uαJ

do not necessarily. This

is circumvent by supposing that, if |J | < k, then∂Φα

J

∂uαK= 0 for any |K| = k. That

is the case when the constraint submanifold C has no constraint of higher order, i.e.C = π−1

k,k−1(πk,k−1(C)), or, more generally, when C bers by πk,k−1 over its image.Taking this into account, we expand the previous equation (4.137), obtaining then

constrained equations of dynamics freed of the Lagrange multipliers

∑I+1i=J

pIiα∂Φα

J

∂uα=

∂LC

∂uα−B j

αj −BJjαj

∂ΦαJ

∂uα; (4.138)

∑I+1i=J

pIiα +∑

I+1i=J

pIiα∂Φα

J

∂uαJ

=∂LC

∂uαJ

−BJjαj −B

Jjαj

∂ΦαJ

∂uαJ

, with |J | = 1, . . . , k − 1;(4.139)

∑I+1i=K

pIiα +∑

I+1i=K

pIiα∂Φα

K

∂uαK

=∂LC

∂uαK

, with |K| = k; (4.140)

where ∂LC

∂uαJ

= ∂L∂uα

J

+ ∂L∂uα

J

∂ΦαJ

∂uαJ

, being LC = L i : C −→ ΛmM the restricted Lagrangian.

We are now in disposition to dene the submanifold W C2 along to which solutions of

the constrained dynamical equation (4.128) exist,

W C2 =

w ∈ W C

0 : (4.140)

=

w ∈ W :

uαJ

= ΦαJ(xi, uα

J)

p = L(xi, uαJ)− pIiα uαI+1i∑I+1i=K

pIiα +∑

I+1i=K

pIiα∂Φα

K

∂uαK

=∂LC

∂uαK

(4.141)


Tangency conditions on X with respect to W C2 will give us the constrained equations

of tangency

AαJj

=∂Φα

J

∂xj+ Aα

Jj

∂ΦαJ

∂uαJ

, (4.142)

Cj =∂LC

∂xj+

k−1∑|J |=0

uαJ+1j

∂LC∂uα

J

−∑

I+1i=J

pIiα

(4.143)

−k−1∑|J |=0

uαJ+1j

∑I+1i=J

pIiα −∑|K|=k

∂ΦαK

∂xj

∑I+1i=K

pIiα −k−1∑|I|=0

uαI+1iBIiαj,

∑I+1i=K

BIiαj =

∂2LC

∂xj∂uαK

−∑

I+1i=K

pIiα∂2Φα

K

∂xj∂uαK

(4.144)

+ AβJj

∂2LC

∂uβJ∂uα

K

−∑

I+1i=K

pIiα∂2Φα

K

∂uβJ∂uα

K

−∑I+1i=K

BIiαj

∂ΦαK

∂uαK

, |K| = k.

Proposition 4.58. Let ΩC2 be the pullback of the premultisymplectic form ΩH to W C2 by

the natural inclusion i : W C2 → W , that is ΩC2 = i∗(ΩH). Suppose that m = dimM > 1,

then the (m + 1)-form ΩC2 is multisymplectic if and only if L is regular along W C2 , i.e. if

and only if the matrix ∂2LC

∂uβR∂uα

K

−∑

I+1i=K

pIiα∂2Φα

K

∂uβR∂uα

K

|R|=|K|=k

(4.145)

is non-degenerate along W C2 .

Proof. First of all, let us make some considerations. By denition, ΩC2 is multisymplecticwhenever ΩC2 has trivial kernel, that is,

if v ∈ TW2, ivΩC2 = 0 ⇐⇒ v = 0 .


if v ∈ i∗(TW2), ivΩH|i∗(TW2) = 0 ⇐⇒ v = 0 .

Let v ∈ TW be a tangent vector whose coecients in an adapted basis are given by

v = γi∂

∂xi+ AαJ

∂

∂uαJ+BIi

α

∂

∂pIiα+ C

∂

∂p.

Using the expression (4.51), we may compute the contraction of ΩH by v,

ivΩH =−BIiα duαI ∧ dm−1xi + AαI dpIiα ∧ dm−1xi − γj dpIiα ∧ duαI ∧ dm−2xij

+

(AαI+1i

pIiα +BIiα u

αI+1i− AαJ

∂L

∂uαJ

)dmx

− γj(pIiα duαI+1i

+ uαI+1idpIiα −

∂L

∂uαJduαJ

)∧ dm−1xj.

(4.146)


In addition to this, let us consider a vector v ∈ TW tangent toW2, that is v ∈ i∗(TW2),we then have that

d(uαJ− Φα

J)(v) = 0, d

∑I+1i=K

pIiα +∑

I+1i=K

pIiα∂Φα

K

∂uαK

− ∂LC

∂uαK

(v) = 0 and dH(v) = 0,

which leads us to the following relations for the coecients of v:

AαJ

= γj∂Φα

J

∂xj+ Aα

J

∂ΦαJ

∂uαJ

, (4.147)

∑I+1i=K

BIiα = γj

∂2LC

∂xj∂uαK

−∑

I+1i=K

pIiα∂2Φα

K

∂xj∂uαK

(4.148)

+AβJ

∂2LC

∂uβJ∂uα

K

−∑

I+1i=K

pIiα∂2Φα

K

∂uβJ∂uα

K

−∑I+1i=K

BIiα

∂ΦαK

∂uαK

C = γj

∂LC∂xj−∑

I+1i=K

pIiα∂Φα

K

∂xj

(4.149)

+AαJ

∂LC∂uα

J

−∑

I+1i=J

pIiα −∑

I+1i=J

pIiα∂Φα

J

∂uαJ

−BIiα u

αI+1i

.

It is important to note that, even though in all the previous equations (4.146), (4.147),(4.148) and (4.149) explicitly appear A's with multi-index of length k, for such a vectorv ∈ i∗(TW2), the terms associated to these A's cancel out in the development of ivΩH,Equation (4.146), and the third tangency relation (4.149). Thus, a tangent vector v ∈i∗(TW2) would kill ΩH if and only if its coecients satisfy the following relations

γj = 0, AαI = 0, BIiα = 0, C = 0,

AαK

= AαK

∂ΦαK

∂uαK

and AβR

∂2LC

∂uβR∂uα

K

−∑

I+1i=K

pIiα∂2Φα

K

∂uβR∂uα

K

= 0.

These considerations being made, the assertion is now clear.

Proposition 4.59. . Let σ ∈ ΓπW,M be an integral section of a solution X of theconstrained dynamical equation (4.128). Then, its Lagrangian part σ1 = pr 1σ isholonomic, σ1 = jkφ for some section φ ∈ Γπ, which furthermore satises the constrainedhigher-order Euler-Lagrange equations (4.46).

Proof. If X is locally expressed as in (4.125), we know that it must satisfy the equationsof dynamics (4.130), (4.132) and (4.132), for unknown Lagrange multipliers λµ. If wenote λ′µ = λµ σ and L′ = L − λ′µΨµ, it suces to follow the proof for L′ of Theorem4.50.


Example: Controlled Fluid Mechanics

Here, we study an incompressible uid under control as in [3]. The corresponding equa-tions are the Navier-Stokes one plus the divergence-free condition:

∂v

∂t+∇vv +∇Π = ν∆v + f (4.150)

∇ · v = 0 (4.151)

where the vector eld v is the velocity of the uid, f is the eld of exterior forces actingon the uid, which will be our controls, and the scalar functions Π and ν are the pressureand the viscosity, respectively. In particular, our case of interest is the two dimensionalcase on R2 endowed with the standard metric. If we x global Cartesian coordinates(x, y) on R2 and adapted coordinates (x, y, u, v) on its tangent TR2 = R4, the previousequations become

ut + u · ux + v · uy + ∂xΠ = ν · (uxx + uyy) + F (4.152)

vt + u · vx + v · vy + ∂yΠ = ν · (vxx + vyy) +G (4.153)

ux + vy = 0 (4.154)

where, with some abuse of notation, v(t, x, y) = (u, v) and f = (F,G).We therefore look for time-dependent vector elds v = (u, v) on R2 that satisfy the

Navier-Stokes equations (4.152) and (4.153) for a prescribed control f = (F,G) andsubmitted to the free divergence condition (4.154). Moreover, we look for such vectorelds v = (u, v) that are in addition optimal in the controls for the integral action

AL(v, R) =1

2

∫R

‖f‖2 dt ∧ dx ∧ dy. (4.155)

In order to apply the development of the present jet bundle framework, all of this isrestated in the following way: We set a ber bundle π : E −→M by puttingM = R×R2,E = R× TR2 and π = (pr 1, prR2). We x global adapted coordinates (t, x, y, u, v) on E,which induce the corresponding global adapted coordinates on Jkπ and Jkπ†. Besides,we choose the volume form η on M to be dt ∧ dx ∧ dy. Thus, the Lagrangian functionL : J2π → R is nothing else but

L =1

2(F 2 +G2),

where we obtain F and G as functions on J2π using the equations (4.152) and (4.153).To make the reading easier, we change slightly the coordinate notation of jet bundles

to t in this example: The coordinate velocities associated to u and v will still belabeled u and v, respectively, with symmetric subindexes (as in the original equations);the coordinate momenta associated to u and v will now be labeled p and q, respectively,with non-symmetric subindexes. Finally and as we will focus on the equations of dynamics(4.138), (4.139) and (4.140), the coecients in the local expression (4.125) of a multivectorX associated to the coordinate momenta p and q will be labeled B and D, respectively.

Example 4.60 (The Euler equation). We will rst suppose that the uid is Eulerian, thatis, it has null viscosity. In this case, the Lagrangian function L = (F 2 +G2)/2 associated


to the integral action (4.155) is of rst order when the Euler equations, (4.152) and(4.153) with ν = 0, are taken into account. In J1π, we consider the divergence-freeconstraint submanifold C = z ∈ J1π : ux + uy = 0, which introduces a single Lagrangemultiplier λ.

Proceeding with the theoretical machinery, we compute the bottom level equations ofdynamics corresponding to those of (4.130)

0 = ux · F + vx ·G− (Btt +Bx

x +Byy )

0 = uy · F + vy ·G− (Dtt +Dx

x +Dyy)

and the top level equations of dynamics (there are no middle ones) corresponding to thoseof (4.132)

pt =F qt =G

px =u · F − λ qx =u ·Gpy = v · F qy = v ·G− λ

We can dispose of the only Lagrange multiplier λ by putting

px − qy = u · F − v ·G,

what denes W C1 together with the top level equations of dynamics with no Lagrange

multiplier.From here, we may compute also the constrained Euler-Lagrange equations (4.46) for

this problem, which are

dtF + u · dxF + v · dyF + vy · F − vx ·G = ∂xλ

dtG+ u · dxG+ v · dyG+ ux ·G− uy · F = ∂yλ

where d∗ = dd∗ .

Finally, we note that L is not regular along W C2 since the square matrix, that corre-

spond to (4.145), 1 u v 0 0u u2 + v2 u · v −v −u · vv u · v v2 0 00 −v 0 1 u0 −u · v 0 u u2

has obviously rank 2. Here we have used as ux as independent (check) coordinate andvy as dependent (hat) coordinate.

Example 4.61 (The Navier-Stokes equation). Now, we tackle the full problem of theNavier-Stokes equations. In this case, the Lagrangian function L = (F 2 + G2)/2 is ofsecond order. In J2π, we consider the constraint submanifold

C =z ∈ J2π : ux + uy = 0, utx + vty = 0, uxx + vxy = 0, uxy + vyy = 0

which comes from the rst order constraint (4.151), free divergence, and its consequencesto second order (see Remark 4.22). These constraints introduce for Lagrange multiplierλ, λt, λx and λy that are associated to them respectively.


We now proceed like in the previous example by computing the equations of dynamics.In rst place, we have the bottom level ones corresponding to those of (4.130)

0 = ux · F + vx ·G− (Btt +Bx

x +Byy )

0 = uy · F + vy ·G− (Dtt +Dx

x +Dyy)

Note that they are formally the same as before. In second place, the mid level equationscorresponding to those of (4.131)

pt =F − (Bttt +Btx

x +Btyy ) qt =G− (Dtt

t +Dtxx +Dty

y )

px =u · F − (Bxtt +Bxx

x +Bxyy ) + λ qx =u ·G− (Dxt

t +Dxxx +Dxy

y )

py = v · F − (Bytt +Byx

x +Byyy ) qy = v ·G− (Dyt

t +Dyxx +Dyy

y ) + λ

Note that formally they also coincide with the top level ones of the previous example butfor the coecients that now appear in them. And in third place, the top level equationscorresponding to those of (4.132)

ptt = 0 qtt = 0

pxx = − ν · F − λx qxx = − ν ·Gpyy = − ν · F qyy = − ν ·G− λy

ptx + pxt = − λt qtx + qxt = 0

pty + pyt = 0 qty + qyt = − λtpxy + pyx = − λy qxy + qyx = − λx

We can again get rid easily of the Lagrange multipliers by putting

ptx + pxt = qty + qyt pxx + ν · F = qxy + qyx pxy + pyx = qyy + ν ·G

what denes W C1 together with the top level equations of dynamics with no Lagrange

multiplier.From here, we may compute also the constrained Euler-Lagrange equations (4.46) for

this problem, which are

2∂2txλt + ∂2

xxλx + 2∂2xyλy − ∂xλ = ∂2

xxν · F + 2∂xν · dxF + ν · d2xxF +

+∂2yyν · F + 2∂yν · dyF + ν · d2

yyF −− dtF − u · dxF − v · dyF − vy · F + vx ·G

2∂2tyλt + 2∂2

xyλx + ∂2yyλy − ∂yλ = ∂2

xxν ·G+ 2∂xν · dxG+ ν · d2xxG+

+∂2yyν ·G+ 2∂yν · dyG+ ν · d2

yyG−− dtG− u · dxG− v · dyG− ux ·G+ uy · F

As before, the Lagrangian is not regular along W C2 , what seems to be clear if we observe

that L is highly non-degenerate: It depends only on 4 of the 12 coordinates of secondorder. It is worthless to show its Hessian, even though it is interesting to say that it isnull only when ν is.


4.2.5 Hamilton-Pontryagin Principle

We next show how the higher-order Euler-Lagrange equations for unconstrained systemscan be derived from a Hamilton-Pontryagin principle (see [156]).

Denition 4.62. Let L : Jkπ −→ ΛmM be a Lagrangian density. The associated(extended) Hamiltonian-Pontryagin action is the map AL : ΓπW,M ×K → R given by

AL(σ,R) :=

∫R

L σk +⟨σ†k, j

1σk−1

⟩−⟨σ†k, σk

⟩(4.156)


Theorem 4.63. A section σ : M → W of πW,M : W → M is a critical point of theHamiltonian-Pontryagin action AL if and only if σk is holonomic, being σk = jkσ0, andσ satises the local equations

0 =∂L

∂uα− ∂σ j

α

∂xj; (4.157)∑

I+1i=J

σIiα =∂L

∂uαJ− ∂σJjα

∂xj, with |J | = 1, . . . , k − 1; (4.158)

∑I+1i=K

σIiα =∂L

∂uαK, with |K| = k. (4.159)

on M , andσIiα = 0, with |I| = 0, . . . , k − 1. (4.160)

on the boundary ∂M of M , where (xi, uαJ , p, pIiα ) denotes adapted coordinates on W and

σ = (xi, σαJ , σ, σIiα ).

Proof. Given a section σ ∈ ΓπW,M and a compact region R ⊆ M , we have that thevariation of the Hamiltonian-Pontryagin action AL with respect to a vertical variation δσof σ is given by

δALδσ

∣∣∣∣(σ,R)

· δσ =

∫R

δ

δσ

[L(xi, σαJ ) + σIiα

(∂σαI∂xi− σαI+1i

)] ∣∣∣∣σ

· δσ dmx

=

∫R

[∂L

∂uαJδσαJ + δσIiα


)+ σIiα

(∂

∂xiδσαI − δσαI+1i

)]dmx

=

∫R

[∂L

∂uαJδσαJ + δσIiα


)− ∂σIiα

∂xiδσαI − σIiα δσαI+1i

]dmx

+

∫∂R

σIiα δσαI dm−1xi

=

∫R

( ∂L

∂uα− ∂σ j

α

∂xj

)δσα +

k−1∑|J |=1

(∂L

∂uαJ− ∂σJjα

∂xj−∑

I+1i=J

σIiα

)δσαJ

+∑|K|=k

(∂L

∂uαK−

∑I+1i=K

σIiα

)δσαK +


)δσIiα

dmx

+

∫∂R

σIiα δσαI dm−1xi


where (xi, uαJ , p, pIiα ) denotes adapted coordinates on W and σ = (xi, σαJ , σ, σ

Iiα ). We

thus infer that σ is a critical point of AL, i.e. δAL/δσ = 0, if and only if the relations(4.1574.160) are satised and σk = jkσ0, what is derived from the last term of the rstintegrand.

4.2.6 The space of symmetric multimomenta

Lemma 4.64. Let (xi, uα) and (yj, vβ) be adapted coordinates on E, whose domains havea non-empty intersection, and let (xi, uαI , p, p

Iiα ) and (yj, vβJ , q, q

Jjβ ) be the corresponding

induced coordinates on the space of forms Λm2 J

kπ, where 0 ≤ |I|, |J | ≤ k.

1. For any pair of multi-indexes I, J ∈ Nm of length k and any pair of indexes 1 ≤α, β ≤ n, the following holds:

∂vβJ∂uαI

=∑π∈Σk

1

I!

∂vβ

∂uα∂xiπ(1)

∂yj1· · · ∂x

iπ(k)

∂yjk, (4.161)

where Σk denotes the collection of permutations π of k elements and the indexes1 ≤ i1, . . . , ik ≤ m and 1 ≤ j1, . . . , jk ≤ m are such that I = 1i1 + · · · + 1ik andJ = 1j1 + · · ·+ 1jk .

2. For any multi-index I ∈ Nm of length k and any indexes 1 ≤ α ≤ n (and 1 ≤ β ≤ n),the following holds:

∑|J |=k

∂vβJ∂uαI

qJjβ =∑j1,...,jk

Jk!

I!

∂vβ

∂uα∂xi1

∂yj1· · · ∂x

ik

∂yjkqJkjβ , (4.162)

where Jk := 1j1 + · · · + 1jk and the indexes 1 ≤ i1, . . . , ik ≤ m are such thatI = 1i1 + · · ·+ 1ik .

Proof. The rst equation is proven by induction on k. The case k = 0 is trivial thus, letus suppose that the result is true for k−1 ≥ 0 and show that it is also true for k. Thanksto Equation (4.4) and the identity (A.6), we may write

∂vβJ∂uαI

=∑

S+1s=I

∂vβJk−1

∂uαS

∂xs

∂yjk

=k∑s=1

1

I(is)

∂vβJk−1

∂uαIs

∂xis

∂yjk,

where we have used the fact that vβJ only depends on uαI 's of order |I| ≥ |J |, which is inthis case on uαI 's of order k. We therefore have by the hypothesis of induction

∂vβJ∂uαI

=k∑s=1

∑π∈Σk−1

1

I(is)

1

Is!

∂vβ

∂uα∂x

isπ(1)

∂yj1· · · ∂x

isπ(k−1)

∂yjk−1

∂xis

∂yjk

=∑π∈Σk

1

I!

∂vβ

∂uα∂xiπ(1)

∂yj1· · · ∂x

iπ(k)

∂yjk.


The second statement is easily proved using the rst one.∑|J |=k

∂vβJ∂uαI

qJjβ =∑j1,...,jk

Jk!

k!

∂vβJk∂uαI

qJkjβ

=∑π∈Σk

∑j1,...,jk

Jk!

k!

1

I!

∂vβ

∂uα∂xiπ(1)

∂yj1· · · ∂x

iπ(k)

∂yjkqJkjβ .

Now, for each permutation π ∈ Σk, we relabel the indexes js in such a way its subindexess coincide with those of is, i.e.∑

|J |=k

∂vβJ∂uαI

qJjβ =∑π∈Σk

∑j1,...,jk

Jk!

k!

1

I!

∂vβ

∂uα∂xiπ(1)

∂yjπ(1)· · · ∂x

iπ(k)

∂yjπ(k)qJkjβ

=∑π∈Σk

∑j1,...,jk

Jk!

k!

1

I!

∂vβ

∂uα∂xi1

∂yj1· · · ∂x

ik

∂yjkqJkjβ

=∑j1,...,jk

Jk!

k!

k!

I!

∂vβ

∂uα∂xi1

∂yj1· · · ∂x

ik

∂yjkqJkjβ .

Note that in any moment Jk is aected by the relabelling.

Theorem 4.65. Let (xi, uαI , p, pIiα ) be an adapted system of coordinates on Λm

2 Jkπ. The

relation

I! · pIiα = I ′! · pI′i′α , whenever I + 1i = I ′ + 1i′ and |I| = |I ′| = k, (4.163)

is invariant under change of coordinates.

Proof. Consider adapted coordinates (xi, uαI , p, pIiα ) and (yj, vβJ , q, q

Jjβ ) on Λm

2 Jkπ, where

0 ≤ |I|, |J | ≤ k, whose domains have a non-empty intersection. Let pIiα a xed coordinatewhere I ∈ Nm is a multi-index of length k, there must be k integers 1 ≤ i1, . . . , ik ≤ msuch that we have the decomposition I = 1i1 + · · ·+1ik . Using the dual coordinate changeformula (4.29) and the previous Lemma 4.64, we obtain

Jac(x(y))pIiα =∑j

∑|J |=k

∂vβJ∂uαI

qJjβ∂xi

∂yj

=∑j

∑j1,...,jk

Jk!

I!

∂vβ

∂uα∂xi1

∂yj1· · · ∂x

ik

∂yjkqJkjβ .

Let (I, i), (I ′, i′) such that I + 1i = I ′ + 1i′ and |I| = |I ′| = k. Then I = I + 1i′ andI ′ = I + 1i, for some multi-index I of length |I| = k − 1. If I! · pIiα = I! · pI′i′α , by theprevious reasoning, we do have

∑j1,...,jk+1

Jk! ·∂vβ

∂uα∂xi1

∂yj1· · · ∂x

ik−1

∂yjk−1

∂xi′

∂yjk∂xi

∂yjk+1qJkjk+1

β =

=∑

j′1,...,j′k+1

J ′k! ·∂vβ

∂uα∂xi1

∂yj′1· · · ∂x

ik−1

∂yj′k−1

∂xi

∂yj′k

∂xi′

∂yj′k+1

qJ ′kjk+1

β


Note that (∂vβ/∂uα) is regular and ∂vβ/∂uα · ∂uα/∂vβ′ = δββ′ , thus∑j1,...,jk+1

Jk(jk)Jk−1! · ∂xi1

∂yj1· · · ∂x

ik−1

∂yjk−1

∂xi′

∂yjk∂xi

∂yjk+1qJkjk+1

β =

=∑

j′1,...,j′k+1

J ′k(j′k)J

′k−1! · ∂x

i1

∂yj′1· · · ∂x

ik−1

∂yj′k−1

∂xi

∂yj′k

∂xi′

∂yj′k+1

qJ ′kjk+1

β

As (∂xi/∂yj) is also regular, we have

∑jk,jk+1

(Jk−1(jk) + 1)Jk−1! · ∂xi′

∂yjkqJkjk+1

β

∂xi

∂yjk+1=

=∑j′k,j′k+1

(Jk−1(j′k) + 1)Jk−1! · ∂xi

∂yj′k

qJ ′kjk+1

β

∂xi′

∂yj′k+1

,

thus(Jk−1(j′) + 1)Jk−1!q

Jk−1+1j′j

β = (Jk−1(j) + 1)Jk−1!qJk−1+1jj

′

β .

Which is equivalent toJ !qJjβ = J ′!qJ

′j′

β ,

whenever J + 1j = J ′ + 1j and |J | = |J ′| = k.

Corollary 4.66. The space of (k + 1)-symmetric multimomenta

Jk+1π‡ := ω ∈ Λm2 J

kπ : I! · pIiα = I ′! · pI′i′α , I + 1i = I ′ + 1i′ , |I| = |I ′| = k (4.164)

is an embedded submanifold of Jk+1π†. A system of adapted coordinates (xi, uα) on Einduces coordinates (xi, uαI , p, p

I′iα , p

Kα ) on Jk+1π‡, where 0 ≤ |I ′| < |I| ≤ |k| and |K| =

k + 1. The natural embedding Jk+1π‡ → Jk+1π† is then given in coordinates by pIiα =pI+1iα /(I(i) + 1), for |I| = k. This manifold is transverse to π†k+1 and therefore bers overJkπ.

Remark 4.67. For the second order case, there is an intrinsic denition of this space thatinvolves the use of the semi-holonomic jets (see Denition 4.26) and which was presentedby Saunders and Crampin in [140].

Note that the k-symmetric multimomenta space Jkπ‡ coincides with the whole dualJkπ† whenever we are considering a rst order theory (k = 1) or a unidimensional one(m = 1). Thus, in the forthcoming discussion we may assume that we are not in any ofthese cases (k,m ≥ 2).

Remark 4.68. Unfortunately, the restriction I! ·pIiα = I ′! ·pI′i′α , I+1i = I ′+1i′ , is no longerinvariant under a change of coordinates when |I| = |I ′| < k. For instance, if |I| = k − 1,we have that

Jac(x(y))pIiα =k∑

|J |=k−1

∂vβJ∂uαI

qJjβ∂xi

∂yj

=∑|J |=k−1

∂vβJ∂uαI

qJjβ∂xi

∂yj+∑|J |=k

∂vβJ∂uαI

qJjβ∂xi

∂yj.


The rst term will be easily expandable to the form (4.162) and, following the proofof Theorem 4.65, it is invariant. However, this is not true for the second term, whichdepends on the chosen coordinates (see Example 4.70 below).

Example 4.69 (Second order case). Consider the dual space Λm2 J

1π of J2π and let (xi,uα, uαi , p, p

iα, p

Ii) and (yj, vβ, vβj , q, qjβ, q

Jjβ ) denote adapted coordinates on it. As the

multi-indexes I and J have unitary length, we may view them as a regular indexes. Inthis case, the higher momenta transform accordingly to

Jac(x(y))pIiα =∑|J |=1

∂vβJ∂uαI

qJjβ∂xi

∂yj

=∂vβ

∂uα∂xI

∂yJqJjβ

∂xi

∂yj.

Indeed, the relation (4.163) that denes the space of 2-symmetric multimomenta is in-variant,

pIiα = piIα =⇒ qJjβ = qjJβ ,

as stated by Theorem 4.65.

Example 4.70 (Third order case). Consider the dual space Λm2 J

2π of J3π and considerthe induced coordinates from adapted ones (xi, uα) and (yj, vβ) on E. Consider a xedmultimomentum coordinate pIiα where I has length |I| = 2. If we assume that I = 1i′′+1i′ ,then the change of coordinates (4.30) reads

Jac(x(y))p1i′′+1i′ ,iα =

∑j

∑|J |=2

∂vβJ∂uα1i′′+1i′

qJjβ∂xi

∂yj

=∑j,j′,j′′

δj′

j′′ + 1

δi′i′′ + 1

∂vβ

∂uα∂xi

′′

∂yj′′∂xi

′

∂yj′q

1j′′+1j′ ,j

β

∂xi

∂yj.

Which proof that the relation I! ·pIiα = I ′! ·pI′i′α , for I+ 1i = I ′+ 1i′ , is invariant whenever|I| = |I ′| = 2.

Let I now denote a multi-index of length |I| = 1. In this case, the rule (4.30) iswritten

Jac(x(y))pIiα =∑|J |=1

∂vβJ∂uαI

qJjβ∂xi

∂yj+∑|J |=2

∂vβJ∂uαI

qJjβ∂xi

∂yj.

The rst term, may be treated as in the previous example 4.69. The second term is

∑|J |=2

∂vβJ∂uαi′

qJjβ∂xi

∂yj=∑j,j′,j′′

[d

dxi′′∂vβ

∂uα∂xi

′′

∂yj′′∂xi

′

∂yj′∂xi

∂yj+

1

2

∂vβ

∂uα∂2xi

′′

∂yj′′∂yj′∂xi

∂yj

]qJ3β .

From here, we see that the relation pi′iα = pii

′α fails to be coordinate independent in Λm

2 J2π,

while it is in Λm2 J

1π (see Example 4.69 and Remark 4.68 above).

Proposition 4.71. Assume that k,m ≥ 2 and consider the pullback Ωs of the canonicalmultisymplectic form Ω of Λm

2 Jkπ to the space of (k + 1)-symmetric multimomenta. We

have that Ωs is still multisymplectic.


Proof. From the local description (4.34) of Ω, we have that

Ωs = − dp ∧ dmx−k−1∑|I|=0

dpIiα ∧ duαI ∧ dm−1xi −∑|K|=k

dpK+1iα ∧ duαK ∧ dm−1xi (4.165)

= − dp ∧ dmx−k−1∑|I|=0

dpIiα ∧ duαI ∧ dm−1xi −∑

|K+|=k+1

dpK+α ∧

∑K+1i=K+

duαK ∧ dm−1xi.

Let V ∈ TJk+1π‡ be of the form

V = γi∂

∂xi+ AαJ

∂

∂uαJ+BIi

α

∂

∂pIiα+BK+

α

∂

∂pK+α

+ C∂

∂p,

then

iV Ωs = −C dmx−k−1∑|I|=0

BIiα duαI ∧ dm−1xi +

k−1∑|I|=0

AαI dpIiα ∧ dm−1xi

−∑|K|=k

BK+1iα duαK ∧ dm−1xi +

∑|K+|=k+1

dpK+α ∧

∑K+1i=K+

AαK dm−1xi.

We deduce from this expression that Ωs has a trivial kernel (iV Ωs = 0 i V = 0), thusΩs is multisymplectic.

This result turns to be trivial for a rst order theory or a unidimensional one since,as stated earlier, in either cases the space of symmetric multimomenta coincides with thewhole dual space.

Symmetric multimomenta constraints within the Skinner-Rusk formalism

We are now in disposition to introduce the k-symmetric multimomenta within the Skinner-Rusk formalism. Two options are available here: First, we could consider the beredproduct Jkπ ×Jk−1π J

kπ‡ and work directly there following the schema of the Skinner-Rusk formalism presented in Section 4.2.3; The second option is to mimic the constrainedversion of it, presented in Section 4.2.4, but considering the k-symmetric multimomentaconstraints in Jkπ† instead of an arbitrary constraint submanifold of Jkπ. We will stickto the latter.

Let W = Jkπ ×Jk−1π Jkπ† be the mixed space of velocities and momenta and W s =

Jkπ ×Jk−1π Jkπ‡ = π−1

2 (Jkπ‡) be the mixed space of velocities and k-symmetric multi-momenta. There is a natural embedding W s → W which is described in coordinates bypIiα = pI+1i

α /(I(i) + 1), where |I| = k− 1 (see Corollary 4.66). Therefore W s is dened bythe constraints I!pIiα = I ′pI

′i′α , where I + 1i = I ′ + 1i′ and |I| = |I ′| = k − 1. As usual, we

consider in addition the constraint H = 0 that denes the Hamiltonian submanifold W0

ofW . Thus, we will work onW s0 = W s∩W0 rather than onW s. If ΩH = Ω+ dH denotes

the Cartan (m + 1)-form of W associated to a Lagrangian density L : Jkπ → ΛmM , wewrite Ωs

H and Ωs0 for their pullbacks to W s and W s

0 , respectively.In order to be able to use the free coordinates of W , we use again the Proposition

4.56 (more precisely, an adaptation of it) that establishes that the dynamical equation interms of multivectors

iXΩs0 = 0, X ∈ X

md (TW s

0 ),


is equivalent to

ii∗XΩH ∈ T 0W s0 , X ∈ X

md (TW s

0 ),

where T 0W s0 is the annihilator of i∗(TW s

0 ) in TW . To write down in coordinates the lastequation, we rst have to describe properly de set of constraints. For each multi-index Kof length k, we x a pair (IK , iK) where IK is a multi-index of length k−1 and 1 ≤ ik ≤ m

such that IK +1iK = K. The set of constraints is I! ·pIiα = (II+1i)! ·pII+1i

iI+1iα , for arbitrary

pairs (I, i) where I is a multi-index of length k− 1 and 1 ≤ i ≤ m. Note that in this set,for each multi-index K of length k, there is a trivial identity for the xed pair (IK , ik).We therefore look for solutions of the dynamical equation

(−1)miXΩH =∑

(I,i)6=(II+1i,iI+1i

)

|I|=k−1

λαIi

(I! · dpIiα − (II+1i)! · dp

II+1iiI+1i

α

)+ λ dH, (4.166)

where X is a multivector eld tangent along W s0 and the λ's are Lagrange multipliers to

be determined.

If we assume that the locally decomposable m-multivector elds X ∈ X(W ) have theform

X = X1 ∧ · · · ∧Xm =∂

∂xj+ AαJj

∂

∂uαJ+BIi

αj

∂

∂pIiα+ Cj

∂

∂p, (4.167)

expanding the rst member in local coordinates and equating coecients, we obtain:

coes. in dp : 0 =λ;

coes. in dpIiα : AαIi =uαI+1i, |I| = 0, . . . , k − 2;

AαIi + I! · λαIi =uαI+1i, |I| = k − 1, (I, i) 6= (II+1i , iI+1i);

AαIK iK −∑

(I,i)6=(IK ,iK)I+1i=K

IK ! · λαIi =uαK , |K| = k;

coes. in duαJ : B iαi =

∂L

∂uα;

BIiαi =

∂L

∂uαI−∑

J+1j=I

pJjα , |I| = 1, . . . , k − 1;

0 =∂L

∂uαK−

∑J+1j=K

pJjα , |K| = k;

coes. in dxj : AαIiBIiαj − AαIjBIi

αi =

(∂L

∂uαJ−∑

I+1i=J

pIiα

)AαJj + uαI+1i

BIiαj.

To get rid o the Lagrange multipliers that appear in the equations coming from the coe-cients of dpIiα with |I| = k−1, we multiply the corresponding equations by I(i) + 1/|I|+ 1and sum over I + 1i = K. Besides, the last equation turns to be null thanks to the otherones and the tangency equations (see below) which come from the k-symmetric restriction


of the multimomenta. So, we do have

AαIi = uαI+1i, with |I| = 0, . . . , k − 2, i = 1, . . . ,m; (4.168)∑

I+1i=K

I(i) + 1

|I|+ 1AαIi = uαK , with |K| = k; (4.169)

0 =∂L

∂uα−B j

αj; (4.170)∑I+1i=J

pI,iα =∂L

∂uαJ−BJj

αj , with |J | = 1, . . . , k − 1; (4.171)

∑I+1i=K

pI,iα =∂L

∂uαK, with |K| = k. (4.172)

Furthermore, we have the tangency conditions

I! ·BIiαj = I ′! ·BI′i′

αj , I + 1i = I ′ + 1i′ , |I| = |I ′| = k − 1; (4.173)∑I+1i=K

BIiαj =

∂2L

∂xj∂uαK+

k−1∑|I|=0

AβIj∂2L

∂uβI ∂uαK

+∑|J |=k

AβJj∂2L

∂uβJ∂uαK

; (4.174)

Cj =∂L

∂xj+ AαJj

∂L

∂uαJ− AαI+1ij

pI,iα −BIiαju

αI+1i

. (4.175)

with respect to the k-symmetry restriction, the equation (4.172) and the zero-level set ofH, respectively. Note that, the Lagrange multipliers are hidden in Equation (4.174) and(4.175) through the coecients AαJj of X of degree k − 1.

Remark 4.72. We have obtained the same equations than in the free case, cf. equa-tions (4.644.67), but with a slight dierence in the highest order equations of holonomy(4.169). What does that imply? An integral section σ ∈ ΓπW,M of a solution X of thedynamical equation (4.166) will no longer be holonomic (at order k) as happens in thefree case, cf. Proposition 4.50 and we will have to require it.

Proposition 4.73. Given a solution X ∈ Xmd (i∗(TW

s0 )) of the dynamical equation

iXΩH ∈ T 0W s0 ,

let σ ∈ ΓπW,M be an integral section of X and denote its Lagrangian part σk = pr 1 σ.If j1(πk,k−1 σk) = σk, then σk is holonomic, i.e. σk = jkφ, and σ0 = πk,0 σk satisesthe higher order Euler-Lagrange equations.

Proof. The hypotesis σk = j1(πk,k−1 σk) directly implies that σ is holonomic, i.e. σk =jk(σ0) (and that the Lagrange multipliers are null along the image of σ). The rest of theproof is the same than the one of Proposition 4.50 (note that equations (4.170), (4.171)and (4.172) coincide with (4.65), (4.66) and (4.67)).

This result ensures that, even with the addition of the k-symmetric multimomentaconstraints, the holonomic integral sections of a solution of the dynamical equation arestill solutions of the Euler-Lagrange equations. Furthermore, there is an improvementwith respect to the free case, Section 4.2.3. If we consider the system of linear equationsin terms of coecients B's with multi-indexes of length k − 1, the highest one, given by


Equation (4.171), (4.173) and (4.174), then the system is overdetermined in oposition tothe free case (see Proposition 4.46). This is because now we have added the tangency con-dition with respect to the k-symmetry, Equation (4.173). If we put BI+1i

αj = BIiαj/(I(i)+1),

for |I| = k−1, which is well dened thanks to (4.173), then Equation (4.171) and (4.174)is rewritten to

∑I+1i=J

pI,iα =∂L

∂uαJ−BJ+1j

αj , with |J | = k − 1; (4.176)

BKαj =

∂2L

∂xj∂uαK+

k−1∑|I|=0

AβIj∂2L

∂uβI ∂uαK

+∑|J |=k

AβJj∂2L

∂uβJ∂uαK

, with |K| = k.(4.177)

Now, the new unknowns BKαj are explecitely given in Equation (4.177). Thus, for xed

coecients AαKj with |K| = k, we may consider Equation (4.176) as an extra constrainton W . Tangency conditions on it will then give conditions on the B's of order k− 2 but,since there are no (k−1)-symmetric constraints on the multimomenta (see Remark 4.68),we have to deal again with an undeterminacy on the coecients of a solution X of thedynamical equation.

Let us recover some examples to clarify this.

Example 4.74 (First order Lagrangian as second order). In Example 4.74, we set up a rstorder Lagrangian L : J1π → ΛmM as a second order one L : J2π → ΛmM by puttingL = L π2,1. We saw that, we cannot go pass the rst constraint manifold even thoughL is completely degenerate (in the second order sense). The space of solutions X of thesecond order dynamical equation is too big and the natural rst order solutions cannotbeen determined from it since the system of linear equations of the coecients B of X isunderdetermined.

We considered the rst and second order velocity-momenta mixed spaces W 1 =J1π ×E J1π† and W 2 = J2π ×J1π J

2π†, together with the premultisymplectic formsΩH and ΩH, where H and H are the corresponding Hamiltonian functions associatedto the Lagrangians L and L. Adapted coordinates are denoted (xi, u, ui, p, p

i) and(xi, u, ui, uK , p, p

i, pij) (with |K| = 2) on W 1 and W 2, respectively. For the sake ofsimplicity, we assume that the bers of π : E →M have dimension n = 1.

W 2 //

''PPPPPPPPPPPPPP

W 1

''OOOOOOOOOOOOOO

J2π

π2,1 //

L

((PPPPPPPPPPPPP J1π //

π1

''PPPPPPPPPPPPP

L

E

π

ΛmM M

Figure 4.7: The 1st and 2nd order Lagrangian settings

If the multivector eld X ∈ Xmd (W 1) and X ∈ Xm

d (W 2), solutions of the respective


(free) dynamical equations iXΩH = 0 and iXΩH = 0, have the form

X =m∧j=1

(∂

∂xj+ Aj

∂

∂u+ Aij

∂

∂ui+Bi

j

∂

∂pi+ Cj

∂

∂p

),

X =m∧j=1

(∂

∂xj+ Aj

∂

∂u+ Aij

∂

∂ui+ AKj

∂

∂uK+ Bi

j

∂

∂pi+ Bki

j

∂

∂pki+ Cj

∂

∂p

),

where |K| = 2. We then obtain the relations

Ai = ui,

0 =∂L

∂u−Bj

j ,

pi =∂L

∂ui, (4.178)

for (W 1,ΩH, X); and

Ai = ui,

Aij = u1i+1j ,

0 =∂L

∂u− Bj

j ,

pi =∂L

∂ui− Bij

j ,

pij + pji = (1i + 1j)! ·∂L

∂u1i+1j

= 0, (4.179)

for (W 2,ΩH, X). Equations (4.178) and (4.179), together with H = 0 and H = 0, denethe corresponding submanifolds W 1

1 and W 21 of W 1 and W 2. The tangency condition to

(4.179) isBijk + Bji

k = 0,

which is not enough to overdetermine the B's of highest order.We therefore introduce the 2-symmetric multimomentum constraint pij = pji in W 2

and denote the resulting submanifold W 2,s. If we use adapted coordinates (xi, u, ui, uK ,p, pi, pK) (with |K| = 2) on W 2,s, then the embedding is given by pij = p1i+1j . Now, asolution X ∈ Xm

d (W 2) of iXΩH = 0 along W 2,s is governed by

Ai = ui,

Aij + Aji = u1i+1j ,

0 =∂L

∂u− Bj

j ,

pi =∂L

∂ui− Bij

j ,

pij + pji = (1i + 1j)! ·∂L

∂u1i+1j

= 0.

This, together with the 2-symmetric multimomentum constraint pij = pji and the tan-gency contitions

Bijk + Bji

k = 0 and Bijk = Bji

k ,


reduces the previous system to

Ai = ui,

Aij + Aji = u1i+1j ,

0 =∂L

∂u− Bj

j ,

pi =∂L

∂ui,

pij = 0,

Bijk = 0,

which is precisely the rst order one (if we ignore the second order terms).If we compare this example with Example 4.55), we have again that the Euler-

Lagrange equations appear as a constraint at the second step of the reduction algorithm(combine the tangency condition to pi = ∂L/∂ui with 0 = ∂L/∂u−Bj

j ). But, in this case,and in contrast to the free setting, now it manages to detect if a second order Lagrangianis actually a rst order one.

Example 4.75 (The second order case). Given a second order Lagrangian L : J2π → ΛmM ,let H : W = J2π×J1π J

2π† → ΛmM be the associated Hamiltonian and let ΩH = Ω− dHdenote the Cartan (m + 1)-form. If we consider the dynamical equation iXΩH = 0 inthe space of mixed velocities and 2-symmetric momentaW s (dened by pijα = pjiα ) insteadof along W s, then a solution X ∈ Xm

d (W s) will be governed by the equations

Aαi = ui,

Aαij + Aαji = uα1i+1j,

0 =∂L

∂uα−B j

αj,

piα =∂L

∂uαi−B1i+1j

αj , (4.180)

pKα =∂L

∂uαK, |K| = 2, (4.181)

where (xi, u, ui, uK , p, pi, pK), |K| = 2, denote adapted coordinates on W s and X has the

form

X =m∧j=1

(∂

∂xj+ Aαj

∂

∂uα+ Aαij

∂

∂uαi+ AαKj

∂

∂uαK+B i

αj

∂

∂piα+BK

αj

∂

∂pKα+ Cj

∂

∂p

).

The tangency condition to Equation (4.181) explicitly gives the coecients BKαj, |K| = 2,

of X. Thus, Equation (4.180) is a space constraint from which we may determine thecoecients B i

αj of X. Moreover, if L was degenerate, then further constraints would bedetermined, so reducing the space of possible solutions.

So far, we have seen that the introduction of the k-symmetric momentum constraintsnot only removes the ambiguity in the simple case of a 1st order Lagrangian viewed froma 2nd order setting, but also the full general problem within the 2nd order setting. Allthis ambiguity was one of the reasons why it was not possible to dene a Legendre trans-form nor a Poincaré-Cartan form in higher-order eld theories, problem of furthermostimportance. Having removed this ambiguity, is it possible now to dene such objects?


Given a solution X ∈ Xmd (W s) of the dynamical equation iXΩH = 0, let σ ∈ ΓπW s,M

be a holonomic integral section of X, meaing that its Lagrangian part σk = pr 1 σ isholonomic. Then,

BKαj σ =

∂σKα∂xj

=∂

∂xj

(∂L

∂uαK σk

)=

(d

dxj∂L

∂uαK

) j1σk.

We therefore dene the 2nd order (extended) Legendre transform as the bered mapLegL : J3π → J2π‡ locally given by

pKα =∂L

∂uαK, |K| = 2, (4.182)

piα =∂L

∂uαi− d

dxj∂L

∂uα1i+1j

, (4.183)

p = L− uαi ·

(∂L

∂uαi− d

dxj∂L

∂uα1i+1j

)− uαK ·

∂L

∂uαK. (4.184)

The Poincaré-Cartan form is then the (m+ 1)-form ΩL along π3,2 locally given by

ΩL =− d

(L− uαi ·

(∂L

∂uαi− d

dxj∂L

∂uα1i+1j

)− uαK ·

∂L

∂uαK

)∧ dmx

− d

(∂L

∂uαi− d

dxj∂L

∂uα1i+1j

)∧ duα ∧ dm−1xi

− d

(∂L

∂uα1i+1j

)∧ duαj ∧ dm−1xi.

(4.185)

For a similar approach, the paper [140] by Saunders and Crampin is strongly recom-mended.

Example 4.76 (The third order case). In this example, we are going to see that theimprovements we got in the second order case by introducing the 2-symmetric momentumconstraints are only partial for the third order case. We x a third order LagrangianL : J3π → ΛmM and look for solutions X ∈ Xm

d (W ), where W = J3π ×J2π J3π†, of

the dynamical equation iXΩH ∈ T 0W s0 , where W

s is the Hamiltonian mixed space ofvelocities and 3-symmetric momenta given by H = 0 and I! · pIiα = J ! · pJjα , for I +1i = J + 1j and |I| = |J | = 2. Recall that, as usual, we denote adapted coordinateson W by (xi, uα, uαi , u

αI , u

αK , p, p

iα, p

i′iα , p

Iiα ), with |I| = 2 and |K| = 3. Thus, we take

coordinates (xi, uα, uαi , uαI , u

αK , p, p

iα, p

i′iα , p

Kα ), with |I| = 2 and |K| = 3, on W s such that

the embedding W s → W is given by pIiα = pI+1iα /(I(i) + 1).

If X ∈ Xmd (W s) has the form

X =m∧j=1

(∂

∂xj+ Aαj

∂

∂uα+ Aαij

∂

∂uαi+ AαIj

∂

∂uαK+ AαKj

∂

∂uαK

+B iαj

∂

∂piα+Bi′i

αj

∂

∂pi′iα+BIi

αj

∂

∂pIiα+ Cj

∂

∂p

),


in order to be a solution of the dynamical equations in W s, its coecients must satisfythe following relations

Aαi = ui,

Aαij = uα1i+1j,∑

J+1j=I

AαJj = uαI , |I| = 2,

0 =∂L

∂uα−B j

αj,

piα =∂L

∂uαi−Bij

αj,∑1i′+1i=I

pi′iα =

∂L

∂uαI−BI+1j

αj , |I| = 2, (4.186)

pKα =∂L

∂uαK, |K| = 3. (4.187)

Tangency conditions on Equation (4.187) gives explicitly all the coecients BKαj, with

|K| = 3. This turns Equation (4.186) into a space constraint in W s; however, tangencyconditions on it do not give enough conditions on the coecients Bi′i

αj to determinedthem, like in the free second order case. In general, the top level coecients BK

αj areoverdetermined inducing a new space constraint on W s, but the subsequent coecientsBIiαj (of order k − 1) with |I| = k − 2 are always undetermined, unless k = 2.This example is of furthermost importance since it is the key step to solve the ambi-

guity that exists in the solutions of the dynamical equation for higher order eld theories.Moreover, to solve or describe this ambiguity will also do it for the denition of thehigher-order Legendre transform and, consequently, higher-order Poincaré-Cartan form.

Chapter 5

Conclusions and future work

As for conclusion, I summarize the main results obtained in this memory.

• First, we have given a description without ambiguity of the higher-order classicaleld theory within a formulation of Skinner and Rusk type, which has permittedto dene a premultisymplectic form and a unique Hamiltonian function; and inconsequence a global and unique formulation of the dynamics. This part of thetreatise has been published in Journal of Physics A: Mathematical and TheoreticalVol. 42 (2009).

• Secondly, we have developed the previous work and exposed an intrinsic formulationof the variational problem equations subjected to constraints dependent on higherorder partial derivatives of the elds with respect to the base coordinates. As a studycase, we have apply this theory to optimal control systems of partial dierentialequations. This results are gathered in the proceedings of dierent congresses: 18thInternational Fall Workshop on Geometry and Physics and Variational Integratosin Nonholonomic and Vakonomic Mechanics; and in a paper that has to appear inthe Journal of Physics A: Mathematical and Theoretical.

• Finally, we have given an important step in order to answer the inherent ambiguityof the Hamiltonian formulation. This work has proven to dene univocally theHamiltonian formulation of classical eld theories of second order; specically, wehave successfully established a space of momenta in which the reduction algorithmdoes not stop and continues giving the subsequent steps.

Besides, also some results have been obtained in continuum mechanics with of applyingthe developed work in classical eld theory to it (see Section 5.4 below).

• Within the theory of constitutive equations of material, a new denition has beengiven for materials know as functionally grade media thanks to their inherent prop-erties. This denition has been proven to generalize the classical one, which hasbeen published in the proceedings of the XVI International Fall Workshop on Ge-ometry and Physics and in the International Journal of Geometric Methods inModern Physics.

In particular, this results are given by Equation (4.52), Proposition 4.46, Proposition4.48, Theorem 4.50, Example 4.55, Equation (4.124), Proposition 4.58, Theorem 4.59,

119

120 CHAPTER 5. CONCLUSIONS AND FUTURE WORK

Theorem 4.63, Theorem 4.66, Theorem 4.73, Examples 4.74 and 4.75, Denition 5.15 andTheorems 5.20 and 5.22.

These results haven been published in

- C. M. Campos, M. Epstein y M. de León, Functionally graded media. Int. J.Geom. Methods Mod. Phys. 5 (2008), no. 3, 431-455.

- C. M. Campos y M. de León, Functionally graded media. Proceedings of the"XVI International Fall Workshop on Geometry and Physics" (2007)

- C. M. Campos, M. de León, D. Martín de Diego and J. Vankerschaver,Unambiguous formalism for higher order Lagrangian eld theories. J. Phys. A:Math. Theor. 42 (2009) 475207 (24pp)

- C. M. Campos, Vakonomic Constraints in Higher-Order Classical Field Theory.Proceedings of the "XVIII International Fall Workshop on Geometry and Physics"(2010)

- C. M. Campos, Higher-Order Field Theory with Constraints. To appear in Rev.R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM

- C. M. Campos, M. de León and D. Martín de Diego, Constrained VariationalCalculus for Higher Order Classical Field Theories. To appear in J. Phys. A:Math. Theor.

The geometrical framework of the developed eld theory is already prepared for itsapplication to dierent lines of research, for instance: continuum mechanics, media withmicrostructure, multisymplectic integrators in higher-order eld theory with or withoutconstraints, etc. I debrief some of them in the following sections.

5.1 Geometric integrators for higher-order eld theo-

ries

During the last years, there was a great interest in developing of geometric integrators formechanical systems using a discrete variational principle (see [123] and references therein).In particular, this eort has been concentrated for the case of discrete Lagrangian func-tions Ld on the cartesian product Q × Q of a dierentiable manifold. This cartesianproduct plays the role of a discretized version" of the standard velocity phase space TQ.Applying a natural discrete variational principle and assuming a regularity condition, oneobtains a second order recursion operator Υ : Q×Q −→ Q×Q assigning to each inputpair (q0, q1) the output pair (q1, q2). When the discrete Lagrangian is an approximationof a continuous Lagrangian function (more appropriately, when the discrete Lagrangianapproximates the integral action for Ld) we obtain a numerical integrator which inheritssome of the geometric properties of the continuous Lagrangian (symplecticity, momentumpreservation). Although this type of geometric integrators have been mainly consideredfor conservative systems, the extension to geometric integrators for more involved sit-uations is relatively easy, since, in some sense, many of the constructions mimic thecorresponding ones for the continuous counterpart. In this sense, it has been recently

5.1. GEOMETRIC INTEGRATORS FOR HIGHER-ORDER FIELD THEORIES 121

shown how discrete variational mechanics can include forced or dissipative systems, holo-nomic constraints, explicitely time-dependent systems, frictional contact, nonholonomicconstraints... All these geometric integrators have demonstrated, in worked examples, anexceptionally good longtime behavior and obviously this research is of great interest fornumerical and geometric considerations (see [104, 135]).

These methods have also extended for lagrangian eld theories (see [120] and referencestherein) of order 1. These methods start by discretizing the spacetime M and in manycases it is assumed for simplicity that M = R2, and Y = R2 × Q, where Q is a vectorspace. Typically, it is considered a mesh as a discretized version of M . Remember that amesh X is a discrete subset of R2. For instance, the quadrangular mesh X = hZ× kZ =xi,j = (hi, kj) | (i, j) ∈ Z × Z. In this sense a discrete eld is a map φd : X −→ Q.In the following we will restrict ourselves to quadrangular mesh although it is easilygeneralizable to other types of meshes. Dene the set of squares X 4 whose elements arethe ordered quadruples of the form

i,j = (xi,j, xi+1,j, xi+1,j+1, xi,j+1)

The idea behind these discretizations is that the values of the discrete eld at thevertices of the squares can be used to dene the concept of discrete jet as an approximationof the continuous jet. In the case of a rst order eld theory the discrete jet bundles isdened as

J1dπ = X 4 ×Q4

and a discrete jet is a pair (i,j, [qi,j, qi+1,j, qi+1,j+1, qi,j+1]).For discretizing the theory it can be useful to dene appropriate discretization maps

Φd = J1dπ → J1π as for instance:

φd((i,j, [qi,j, qi+1,j, qi+1,j+1, qi,j+1])

= (xi,j + xi+1,j + xi+1,j+1 + xi,j+1

4,qi,j + qi+1,j + qi+1,j+1 + qi,j+1

4, V1, V2)

where

V1 =1

2

(qi+1,j − qi,j

k+qi+1,j+1 − qi,j+1

h

),

V2 =1

2

(qi,j+1 − qi,j

h+qi+1,j+1 − qi+1,j

k

),

which are considered as an approximation of the partial derivatives of the eld.Then, given a lagrangian L : J1π → R, we dene the discrete lagrangian Ld : J1

dπ → Rby Ld = hkΦ∗d.

The discrete eld equations are deduced extremizing an appropriate discrete sum. Inthis particular case, the discrete eld equations are (see [120]:

0 = D1Ld((i,j, [qi,j, qi+1,j, qi+1,j+1, qi,j+1])

+D2Ld((i,j−1, [qi,j−1, qi+1,j−1, qi+1,j, qi,j])

+D3Ld((i−1,j, [qi−1,j, qi,j, qi,j+1, qi−1,j+1])

+D4Ld((i−1,j−1, [qi−1,j−1, qi,j−1, qi,j, qi−1,j]).


Of course we can generalize these methods for higher order eld theories adding dis-cretizations of the higher-order derivatives (the second order case is already studied in[112]). For instance, for a second-order lagrangian we can consider a discrete Lagrangiandened by

L2d : (X 4 ×X 4)2 ×Q7 → R

where (X 4×X 4)2 are rectangles such that the third right-upper vertex of the rst rectangleis also, the left-bottom vertex of the second one. Now, a discretization of the second-orderderivatives is given, for instance, by:

V11 =qi+1,j − 2qi.j + qi−1,j

2h

V22 =qi,j+1 − 2qi.j + qi,j−1

2k

V12 = V21 =qi+1,j+1 − qi,j+1 − qi,j−1 + qi−1,j−1

2hk

In future research we will study these methods for higher order lagrangian systems in-cluding their geometric preservation properties (multisymplecticity, etc.). Moreover, it ispossible to extend these techniques for the case of Lagrangian systems with constraints(see [15]).

5.2 Space+Time Decomposition

As for the Skinner-Rusk formalism, another framework of interest is the so called space+time decomposition originally developed by Gotay in [96] (see also [21]). This formalismis strongly based on the theory of Cauchy surfaces, in which ones assumes that thereexists a space-like surface in the ambient space that evolves along the time line such thatit covers the whole ambient space. This description allow us to consider any eld theoryin frozen time and then watch it evolve.

To be more precise, let as usual π : E → M be a ber bundle whose bers havedimension n but whose base manifold, which is assumed to be orientable and orientedwith a provided volume form η, has now dimension m + 1. We assume that there existsan m-dimensional manifold X that can be embedded into M . Let ε ∈ Emb(X,M) beone of such embeddings, we view Mε := ε(X) as a Cauchy surface. We now consider theeld theory restricted to Mε, that is me shall consider the ber bundle πε : Eε → Mε,where Eε = EMε = π−1(Mε) and πε = π|Eε . The space of sections Eε = Γπε is called theinstantaneous conguration space at time ε.

In this setting, given a section σ ∈ Eε we have that the tangent space to Eε at σ is

TσEε = v : Mε → V πε | v covers σ ,

and the cotangent space to Eε at σ is

T ∗σ Eε = α : Mε → L(V πε,ΛmMε) | α covers σ ,

where L(V πε,ΛmMε) is the vector bundle over Eε whose ber at u ∈ (Eε)x is the set oflinear maps form VuEε to Λm

xMε. Thus the pairing between the elements of T ∗σ Eε andthose of TσEε is given by the integral expression:

〈α, v〉 =

∫Mε

α(v).

5.3. REDUCTION 123

Furthermore, we can dene a Liouville form on the ε-phase space T ∗Eε in the usualmanner:

θε(α)(V ) =⟨α, TαπEε(V )

⟩.

And, of course, the canonical symplectic form ωε = − dθε.Before we give a Lagrange description of the eld dynamics within this setting, we

must introduce two concepts. First, we consider a slicing of M with section X, that is atime-dependent family of embeddings χ : I × X → M , where I ⊂ R, such that χ is infact a dieomorphism. We dene the generator of χ as the push forward of ∂/∂t, that is

ξχ χ := Tχ

(∂

∂t

).

Secondly, we assert that TEε is isomorphic to the collection of restrictions of holonomicsections of π1 : J1π →M (see [96]).

Now, given a Lagrangian density L : J1π → ΛmM , we dene a Lagrangian functionL : TEε → R in the following way:

Lε,χ(σ) =

∫Mε

iξχL(j1φ),

where j1φ is the holonomic section that corresponds to σ.From here we could proceed in the standard ways but, we remark that we nally

have the three basic elements to follow the Skinner-Rusk formalism: the Lagrangian,the pairing and the canonical form. The goals of this work is to study the space+timedecomposition within the Skinner-Rusk formalism and extend it to higher-order theories.Since the base manifold adds a new data in the picture, the slicing, it could possiblyreduce the ambiguity in the space of solutions.

5.3 Reduction

Among dierent extra structures that the ber bundle π : E →M may carry, of particularinterest is the case when π is a principal ber bundle. In this context and under extraassumptions on the Lagrangian, one may seek for symmetries of the problem or usereduction techniques to eliminate variables and simplify the problem. This is a naturalstep when a dynamic formalism is well established, which is the case of rst order classicaleld theories, and which has already started (see for instance [37, 38, 39]).

The aim of a future work is to study and develop a theory of multisymplectic reductionin higher-order eld theories and, in view of example 4.75, particularly for the second ordercase.

5.4 Continuous Media

The study of the mechanics of continuous media constitutes a non-trivial example of the-ory of classical elds, whose structure and dynamics may be characterized geometrically.Nonetheless, there still is a long way in process to geometrize this study. From the useof Lie algebra to describe the movement of a rigid body, to the modeling of Cosserat


media and liquid crystals by means of principal ber bundles. Before focusing on the dy-namical aspects of a continuum, one should start studying the behavior of a body underinnitesimal deformations in order to understand its internal structure, which is the basisof constitutive theory of materials. In this sense, what follows is the work developed in[28], which is a study of materials that gradually change its behavior from point to point,that is, functionally graded media.

The mechanical response at a point X of a simple (rst-grade) local elastic body Bdepends on the rst derivative F at X ∈ B of the deformation. In other words, B obeysa constitutive law of the form:

W = W (F (X);X) (5.1)

where W measures the strain energy per unit volume. The linear map F (X) is calledthe deformation gradient at X. Of course, there are materials for which the constitutiveequation implies higher order derivatives or even internal variables as it happens withthe so-called Cosserat media or, more generally, media with microstructure, but suchmaterials will not be considered here.

An important problem in Continuum Mechanics is to decide if the body is made ofthe same material at all its points. To handle this question in a proper mathematicalway, one introduces the concept of material isomorphism, that is, a linear isomorphismPXY : TXB −→ TYB such that

W (FPXY ;X) = W (F ;Y )

for all deformation gradients F at Y . Intuitively, this means that we can extract asmall piece of material around X and implant it into Y without any change in themechanical response at Y . If such is the case for all pairs of body points, we say thatthe body B is uniform. This has been the starting point of the work by Noll and Wang[130, 146, 155, 154] in their approach to uniformity and homogeneity.

In this context, a material symmetry at X is nothing but a material automorphismof the tangent space TXB. The collection of all the material symmetries at X formsa group, the material symmetry group G(X) at X. An important consequence of theuniformity property is that the material symmetry groups at two dierent points X andY are conjugate.

A natural question arises: Is there a more general notion that permits to compare thematerial responses at two arbitrary points even if the body does not enjoy uniformity? Ananswer to this question is based on the comparison of the symmetry groups at dierentpoints. Indeed, we say that the body B is unisymmetric if the material symmetry groupsat two dierent points are conjugate, whether or not the points are materially isomor-phic. From the point of view of applications, this kind of body corresponds to certaintypes of the so-called functionally graded materials (FGM for short). The unisymmetryproperty was introduced in [81] with the objective to extend the notion of homogeneityto non-uniform material bodies. Let us recall that the homogeneity of a uniform body isequivalent to the integrability of the associated material G-structure [22, 80]. Roughlyspeaking, this material G-structure is obtained by attaching to each point of B the cor-responding material symmetry group via the choice of a given linear reference at a xedpoint; a change of the linear reference gives a conjugate G-structure. In a more sophis-ticated framework, the set of all material isomorphisms denes a Lie groupoid, which insome sense is a way to deal with all these conjugate G-structures at the same time.

5.4. CONTINUOUS MEDIA 125

In the case of unisymmetric materials the attached group is not the material sym-metry group, but its normalizer within the whole general linear group. This implies amore dicult understanding of the generalized concept of homogeneity associated withunisymmetric materials. The main aim of the present paper is to provide a convenientcharacterization of this homogeneity property. In this sense, this work may be regardedas a continuation and improvement of the results obtained in [81].

The paper is organized as follows. Section B.1 is devoted to a brief introduction togroupoids and Lie groupoids; in particular, we dene the normalizoid of a subgroupoidwithin a groupoid, which is just the generalization of the notion of normalizer in thecontext of groups. An important family of examples is provided by the frame-groupoid,consisting of all the linear isomorphisms between the tangent spaces at all the pointsof a manifold M ; if M is equipped with a Riemannian metric g, one can introduce thenotion of orthonormal groupoid (taking the orthogonal part of the linear isomorphismsgiven by the polar decomposition). If, without necessarily possessing a distinguishedRiemannian metric,M is endowed with a volume form, one obtains the Lie subgroupoid ofunimodular isomorphisms. In Section B.2 we analyze the relations between Lie groupoidsand principal bundles; in particular, we examine the relation between the frame groupoidand G-structures on a manifold M . In Section 5.4.1 we study the concepts of materialsymmetry and material symmetry groups, and in Section 5.4.2 we discuss uniformityand homogeneity. Finally, Section 5.4.3 is devoted to study the case of FGM materials,and the geometric characterization of homogeneity in this case is obtained for both solidand uids.

5.4.1 The Constitutive Equation

In the most general sense (see [119], for instance), a body is a manifold B that can beembedded in a Riemannian manifold (S, g) with the same dimension, the ambient space.Usually, the body B is a simply connected open set of R3 and the ambient space is R3

itself with the standard metric. Each embedding K : B → S is called a congurationand its tangent map TK : TB → TS is called an innitesimal conguration. If we x aconguration K (the reference conguration) and we pick an arbitrary conguration K,then the embedding compositon φ = K K−1 : K(B) ⊂ S → S is considered as a bodydeformation and we call its tangent map TXφ at a point X in B an innitesimal defor-mation or the deformation gradient, usually denoted by F . Since (S, g) is a Riemannianmanifold, we can induce a Riemannian metric on B by the pull-back of g by a referenceconguration K. Since the metric on B depends from a chosen reference conguration,it is not canonical. However, for solid materials, we are able to dene an almost uniquemetric compatible with the material structure, as we will show in section 5.4.2.

Usually, points in the body or in the reference conguration (when they are identied)are denoted by capital letters X, Y , Z, etc., and by small letters x, y, z, etc., in thedeformed conguration. At the moment we have the picture shown at Figure 5.1.

As stated by the principle of determinism, the mechanical and thermal behaviors ofa material or substance are determined by a relation called the constitutive equation. Itdoes not follow directly from physical laws but it is combined with other equations thatdo represent physical laws (the conservation of mass for instance) to solve some physicalproblems, like the ow of a uid in a pipe, or the response of a crystal to an electriceld. In our case of interest, elastic materials, the constitutive equation establishes that,


Figure 5.1: Deformation in a reference conguration.

in a given reference conguration, the Cauchy stress tensor depends only on the materialpoints and on the innitesimal deformations applied on them, that is

σ = σ(FKr , Kr(X)). (5.2)

This relation is simplied in the particular case of hyperelastic materials, for which equa-tion (5.2) becomes

W = W (FKr , Kr(X)). (5.3)

where W is a scalar valued function which measures the stored energy per unit volume.Among other postulates (principle of determinism, principle of local action, principle

of frame-indierence, etc.), it is claimed that a constitutive equation must not dependon the reference conguration. It turns out that equation (5.2) (and (5.3)) now can bewritten in the form

σ = σ(F,X) (W = W (F,X), respectively), (5.4)

where F stands for the tangent map at X of a local conguration (deformation).

Denition 5.1. A material symmetry at a given point X ∈ B is a linear isomorphismP : TXB → TXB such that

σ(F · P,X) = σ(F,X), (5.5)

for any deformation F at X. The set of material symmetries at X ∈ B is denoted byG(X) and it is called the symmetry group of B at X. Given a conguration K, we willdenote by GK(X) the symmetry group G(X) in the conguration K, that is

GK(X) = TXK · G(X) · (TXK)−1. (5.6)

Dierent types of elastic materials are given in terms of their symmetry groups. Forinstance, a point is solid whenever its symmetry group in some reference conguration isa subgroup of the orthogonal group O(3) and, uid whenever the orthogonal group is aproper subgroup of the symmetry group. In [118, 154] it is possible to nd a classication,due to Lie, of the connected Lie subgroups of Sl(3) and their corresponding Lie algebras.

Denition 5.2. Given an elastic material B, let X ∈ B and consider its symmetry groupG(X). If there exists a conguration K such that:

1. GK(X) is a subgroup of the orthogonal group of transformations O(3), then X issaid to be an elastic solid point. If furthermore


Figure 5.2: Material symmetry.

(a) GK(X) = O(3), then we call X a fully isotropic elastic solid point ;

(b) GK(X) is a transverse orthogonal group (a group of rotations which x anaxis), then X is said to be a transversely isotropic elastic solid point ;

(c) GK(X) consists only of the identity element, then X will be a triclinic elasticsolid point ;

2. GK(X) is a subgroup of the unimodular group of transformations U(3) and has theorthogonal group O(3) as a proper subgroup, then X is said to be an elastic uidpoint. If furthermore

(a) GK(X) = Sl(3) then we still call X an elastic uid ; and

(b) GK(X) is a transverse unimodular group (a group of unimodular transforma-tions which x an axis or a group of unimodular transformations which x aplane) then we call X an elastic uid crystal.

The innitesimal conguration TXK or the induced frame z = (TXK)−1 is called anundistorted state of X.

This material classication is pointwise. A body is solid if every point is solid.

5.4.2 Uniformity and Homogeneity

To dene the uniformity of a material, we rst have to give a criterion that establisheswhen two points are made of the same material. To compare their symmetry groups isnot sucient since this is only a qualitative aspect. Indeed, consider two points in arubber band, one point may be relaxed while another point may be under stress. But weare still able to release the stress on the second point and bring it to the same state asthe rst one, and then compare their responses.

Denition 5.3. We say that two points X, Y ∈ B are materially isomorphic, if thereexists a linear isomorphism PXY : TXB → TYB such that

σ(F · PXY , X) = σ(F, Y ), (5.7)

for any deformation F at Y . The linear map PXY is called a material isomorphism.

Even if the denition of material isomorphism and material symmetries are mathe-matically similar, there is an important conceptual dierence. While the symmetry groupof a point characterizes the material behavior of that point, a material isomorphism estab-lishes a relation between two dierent points. In fact, as already pointed out, a material


Figure 5.3: Material isomorphism.

symmetry can be viewed as a material automorphism by identifying X with Y in theabove denition.

Denition 5.4. Given a material body B, the material groupoid is the set of all thematerial isomorphisms and symmetries, that is the set

G(B) = P ∈ Π(B) satisfying Denition 5.3 . (5.8)

It is easy to check that the material groupoid G(B) is actually a groupoid. Further-more, it is a subgroupoid of the frame groupoid Π(B), but note that it is not necessarilya Lie groupoid or even transitive as the frame groupoid. In fact, when all the points ofa body are pairwise related by a material isomorphism, it means that the body consistsonly of one type of material. In this case, it is materially uniform.

Denition 5.5. Given a material body B, we say that it is uniform if the materialgroupoid G(B) is transitive, and smoothly uniform when the material groupoid is a tran-sitive dierential groupoid (and hence a Lie subgroupoid of Π(B)).

A simple but important property of uniform materials is that the groups of materialsymmetries are mutually conjugate by any material isomorphism between the respectivebase points. To be more precise, equation (B.2) reads in terms of elastic bodies:

G(Y ) = P · G(X) · P−1, ∀P ∈ G(B)X,Y , (5.9)

for any pair of materially isomorphic points X, Y ∈ B.When we look a material through dierent congurations, there are prefered states

of the material we want to distinguish: e.g. transversely isotropic solids have a xed axisinvariant under material isomorphisms that we prefer to align with the vertical axis.Such a state may be modelized in an innitesimal conguration by a linear frame z. As wehave just said, in the material paradigm, this frame of reference z has some behaviors thatwill be mainted by material isomorphisms. If we consider the set of all these distinguishedreferences that arise from material transformations of the `reference crystal' (see Figure5.4), then we obtain the so called material G-structure of B. As far as we know, Wangwas the rst to realize that the uniformity of a material can be modeled by a G-structure[154], although this fact was emphasized by Bloom [22]. For deniteness,

Denition 5.6. A material G-structure of a smoothly uniform body B is any of theGz-structures induced by the material groupoid G(B) as shown in Theorem B.17. Thechosen frame of reference z ∈ FB is called the reference crystal.


Figure 5.4: The reference crystal.

Denition 5.7. Given a smoothly uniform body B, a conguration K that induces across-section of a material G-structure will be called uniform. If there exists an atlas(Uα, Kα)α∈A of B of local uniform congurations for a xed material G-structure, thebody B will be said locally homogeneous, and (globally) homogeneous if the body B maybe covered by just one uniform conguration.

The material concept of homogeneity corresponds to the mathematical concept of in-tegrability. By Theorem B.22, a smoothly uniform body B will be locally homogenous ifand only if one (and therefore any) of the associated material G-structures is integrable.Let K a uniform conguration for a particular integrable G-structure G(B) of a homo-geneous elastic material B. If (X, v1, v2, v3) denotes the cross section induced by K, thusthe constitutive equation (5.2) may be written in the form

σ = σ(FK , K(X)) = σ(F ij , x

i), (5.10)

with obvious notation. Now note that, since through K any material isomorphism Pmay be considered as an element of the structure group G, which is clear for materialsymmetries, and since the body B is uniform, we have that

σ(F ij , y

i) = σ(FK , K(Y )) = σ(FK · PK , K(X)) = σ(F ik · P k

j , xi) = σ(F i

j , xi). (5.11)

Thus, we have just proved the following result:

Theorem 5.8. If K is a uniform conguration of a homogeneous elastic body B, theconstitutive equation (5.2) is independent of the material point and invariant under theright action of the structure group G of the G-structure G(B) related to K. Thus,

σ = σ(F ij ) and σ(F i

k · P kj ) = σ(F i

j ) for any P ∈ G. (5.12)

The physical interpretation of this theorem is that points of a homogenous elasticbody B can be put by means of a conguration K in such a manner they are all at thesame state, at least locally. This conguration K is uniform. Even if the material G-structures of a smoothly uniform body B are dierent (but equal via conjugation), theremust be at least one of them in which the structure group G satises a condition of thematerial classication 5.2.

Denition 5.9. Accordingly to Denition 5.2, a smoothly uniform elastic body B is solidor uid, if all the points are solid or uid, respectively. Any of the material G-structuresfor which the structure group fullls the classication is called undistorted.


Uniform Elastic Solids

The following result is due to Wang (cf. [154]). In his paper, Wang denes the materialG-structures from the point of view of atlases, families of cross-sections of the framebundle, instead of our approach through groupoids. These families are the cross-sectionsof the resulting G-structures. When a material is solid, it is possible to endow the bodywith a metric wich is compatible with the material structure. Wang calls such a metrican intrinsic metric.

Theorem 5.10. Let B be a uniform elastic solid material; each undistorted materialG-structure G(M) denes a Riemannian metric g, invariant under material symmetriesand isomorphisms.

Proof. Given a cross-section (U, σ) of a xed undistorted material G-structure G(B), letX ∈ U and dene

gσX(v, w) :=⟨σ(X)−1 · v, σ(X)−1 · w

⟩, ∀X ∈ U,∀v, w ∈ TXB, (5.13)

where 〈 , 〉 is the Euclidean scalar product. Thus, gσ is clearly a smoooth positive denitesymmetric bilinear tensor eld on U , since it is nothing more than the pullback of theEuclidean metric. Let us check that, in this manner, the metric gσ does not depend onthe chosen cross-section (U, σ). Given any other cross-section (V, τ), let X ∈ B be in theintersection of their domains (if not empty, of course), then

gσX(v, w) = 〈σ(X)−1 · v, σ(X)−1 · w〉= 〈Q · τ(X)−1 · v,Q · τ(X)−1 · w〉= 〈τ(X)−1 · v, τ(X)−1 · w〉= gτX(v, w),

(5.14)

where we used the fact that, by hypothesis, Q = σ(X)−1 · τ(X) ∈ G is orthogonal.Now, let P ∈ GX,Y (B) be a material isomorphism; there will exist cross-sections

(U, σ), (V, τ) such that P = τ(Y ) · σ(X)−1. Then, we have

gY (P · v, P · w) = 〈τ(Y )−1 · P · v, τ(Y )−1 · P · w〉= 〈σ(X)−1 · v, σ(X)−1 · w〉= gY (v, w).

(5.15)

The metric we where looking for is just the metric g dened in (5.13).

If we consider the orthogonal groupoid O(B) related to this metric, we have thatthe material groupoid is included in it, G(B) ⊂ O(B). Reciprocally, if B is a smoothlyuniform material such that it can be endowed with a Riemannian metric for which thematerial symmetries and isomorphisms are orthogonal transformations, G(B) ⊂ O(B),then B must be an elastic solid. Thus, elastic solids are completely characterized byRiemannian metrics with the property of being invariant under material symmetries andisomorphisms.

Remark 5.11. Given two material G-structures, G1(B) and G2(B), of a uniform elasticsolid B, we know that they must be related by the right action of a linear isomorphism F ∈Gl(3), that is G2(B) = G1(B) · F . Thus, if G1(B) is undistorted, the G-structure G2(B)will be undistorted if and only if the symmetric part V of the left polar decompositionof F , F = V · R, lies in the centralizer of G1, that is V ∈ C(G1) (cf. [154], proposition11.3). But this does not imply that G1(B) and G2(B) dene the same metric, which istrue only if V = I.


Uniform Elastic Fluids

There are similar results for uids as for solids. In this case, the uid structure inducesvolume forms.

Proposition 5.12. Let B be a uniform uid material, then each undistorted materialG-structure G(B) denes a volume form ρ invariant under material symmetries andisomorphisms.

Proof. Given a cross-section (U, σ) of a xed undistorted material G-structure G(B), letus dene on U the volume form

ρσ = σ∗1 ∧ σ∗2 ∧ σ∗3, (5.16)

where σ∗ denotes the co-frame cross-section of σ, that is σ∗ : U −→ F∗B such thatσ∗i(σj) ≡ δij on U . Let us show that the volume form ρσ does not depend on the chosencross-section (U, σ). In fact, let (U, σ), (V, τ) be two cross-sections with non-empty domainintersection, then for any n vectors v1, . . . , vn ∈ TXB, with X ∈ U ∩ V , we have

ρσ(v1, . . . , vn) = det(vji )

= det((σ−1τ)ki ) · det(vjk)

= ρτ (v1, . . . , vn),

where we have used vi = vjiσi = vji τi, vji = (σ−1τ)ki · v

jk and σ−1τ ∈ U(n). Since the

tangent vectors v1, . . . , vn are arbitrary, ρσ and ρτ coincide on the intersection of theirdomains, U ∩ V . Thus, the volume form given in (5.16) denes locally a volume form ρon the whole material body B.

Let us see how ρ is invariant under material symmetries and isomorphisms. GivenP ∈ GX,Y (B), there must exist cross-sections (U, σ), (V, τ) such that P = τ(Y ) · σ(X)−1.Then, we have

ρ P = (P−1τ)∗1 ∧ (P−1τ)∗2 ∧ (P−1τ)∗3 = σ∗1 ∧ σ∗2 ∧ σ∗3 = ρ, (5.17)

which nishes the proof.

Considering now the induced unimodular groupoid U(B), by the invariance we havethe inclusion G(B) ⊂ U(B) which also characterizes elastic uids.

5.4.3 Unisymmetry and Homosymmetry

As we have seen, the concept of homogeneity must be understood within the frameworkof uniformity. But, there are materials that are not uniform by their very denition, theso called functionally graded materials, or FGM for short. This type of material can bemade by techniques that accomplish a gradual variation of material properties from pointto point: for instance, ceramic-metal composites, used in aeronautics, consist of a platemade of ceramic on one side that continuously change to some metal at the oppositeface. The material properties are also given through a constitutive equation like (5.4).Therefore, we will have a notion of material symmetry and the symmetry groups willbe non-empty as in the case of uniform materials. For a FGM material, the symmetrygroups at two dierent points are still conjugate, accordingly to the following denition.


Denition 5.13. Given a functionally graded material B, let be X, Y ∈ B; we say that alinear map A : TXB −→ TYB is a unisymmetric (material) isomorphism if it conjugatesthe symmetry groups of X and Y , namely,

G(Y ) = A · G(X) · A−1. (5.18)

As for uniform bodies, the material properties of a FGM are now characterized by thecollection of all the possible unisymmetric isomorphisms.

Denition 5.14. Given a functionally graded material B, the set of unisymmetric iso-morphisms, that is the set

N (B) =A ∈ Π(B) : G(Y ) = A · G(X) · A−1

, (5.19)

will be called the FGM material groupoid of B.

Figure 5.5: The FGM material groupoid.

We may now extend the ideas of section 5.4.2 using this new object. Then we obtain:

Denition 5.15. A functionally graded material B will be said unisymmetric if the FGMmaterial groupoid N (B) is transitive and, smoothly unisymmetric if it is a Lie groupoid.

Note that the notion of unisymmetry covers a qualitative aspect in the sense that aunisymmetric FGM is made of only one type of material. For instance, it will be a fullyisotropic solid everywhere or a uid everywhere, but it cannot be a fully iscotropic solidat some point and a uid at another point.

For this groupoid, we also have the associated G-structures.

Denition 5.16. Let B be a smoothly unisymmetric body. Any of the asociated G-strutures Nz(B), with z ∈ FB, will be called a material N-structure. A cross-section ofa material N -structure will be a unisymmetric cross-section and a conguration inducingsuch a cross-section will be a unisymmetric conguration. If for any of the material N -structures there exists a covering by unisymmetric congurations, the body B will be saidlocally homosymmetric, and (globally) homosymmetric if the covering consists of only oneunisymmetric conguration.

As we may see, the homosymmetry property is equivalent to the integrability of anyof the material N -structures. However, there is not an analogue result to Theorem 5.8 forhomosymmetric bodies. Since, even if we have an N -structure and the group structureis the same for any point through any unisymmetric conguration, the symmetry groupsmay be represented by dierent subgroups of N at each point.


Functionally Graded Elastic Solids

Denition 5.17. We will say that a functionally graded elastic material B is a func-tionally graded solid if there is a Riemmanian metric on B invariant under materialsymmetries, that is every point is solid. Furthermore, B will be said

1. fully isotropic if every point is fully isotropic;

2. transversely isotropic if every point is transversely isotropic; and

3. triclinic if every point is triclinic.

The compatible metric is called a material metric.

We have not used the term intrinsic for the material metric, since it does not arisefrom the material structure as for uniform elastic solids (cf. Theorem 5.10). The materialmetric is an extra structures that ensures that the solid points are glued in a solid way.

If B is a FGM solid and we consider the orthonormal cross-sections (U, σ) of theO(3)-structure given by a solid metric, then they must verify:

σ(X)−1 · G(X) · σ(X) ⊆ O(3) ∀X ∈ U ∀(U, σ), (5.20)

σ(X)−1 · τ(X) ∈ O(3) ∀X ∈ U ∩ V ∀(U, σ), (V, τ); (5.21)

where G(X) is the material symmetry group of B at X. In fact, these two conditions arenecessary and sucient to dene a solid metric compatible with the material structureby means of a family of cross-sections of FB.

On the other hand, if we consider another O(3)-structure, giving a second solid metric,the two structures are not a priori related by the right action of a linear isomorphismF ∈ Gl(3). But if they are, then the symmetric part of the polar decomposition of Fmust be spherical, a homothety. This can be interpreted as the material being in bothcases in the same state but the measures of stress, or strain, are performed with dierentscales.

Denition 5.18. A solid FGM B will be said to be relaxable if the O(3)-structuregiven by some solid metric is integrable or, equivalently, if the Riemannian curvature(with respect to this metric) vanishes identically. We then say that the O(3)-structure isrelaxed.

Denition 5.19. We say that a body B is homosymmetrically relaxable if B is an unisym-metric solid material for which there exists a covering Σ of local conguration that areboth, unisymmetric and relaxed congurations.

Let B be a homosymmetrically relaxable elastic solid, then we have these two struc-tures, the unisymmetric and the orthogonal, which are in certain manner interconnected.As B is a solid, intuitively we may perceive that only the orthogonal part of a unisym-metric isomorphism must be important. In what follows, we will explain this fact in moredetail.

A direct consequence of the previous Lemma B.11 and Proposition B.23 is the followingtheorem, which implies a result proved by Epstein and de León [81].


Theorem 5.20. If B is relaxable elastic solid that is also homosymmetric, we have

N (B) = N (B) ∩ O(B), (5.22)

where N (B) consits in the orthogonal part of the isomorphisms of N (B). Therefore, ifNz(B) is a smooth Nz-structure, B will be homosymmetrically relaxable if and only if thereduced material groupoid Nz(B) is integrable (where z ∈ FB is xed).

Let B a relaxable and homosymmetric elastic solid and let g denote the compatiblematerial metric

• If B is fully isotropic, which means the symmetry group G(X) of each point X ∈ Bis equal to the orthogonal group O(TXB, g) itself, then the reduced FGM materialgroupoid N (B) coincides with the orthogonal groupoid O(B).

• If B is triclinic (the only element of the symmetry group is the identity map), theFGM material groupoid N (B) is the full frame groupoid Π(B), and thus N (B) =O(B) as before.

• If B is transversally isotropic, at each point X ∈ B there exists a basis of TXB inwhich the material symmetries g ∈ G(X) may be represented by matrices of theform: 1 0 0

0 cos θ − sin θ0 sin θ cos θ

Thus, for this basis, the normalizer of G(X) is

N (X) =

⟨1 0 00 cos θ − sin θ0 sin θ cos θ

,

α 0 00 β 00 0 β

⟩

where the brackets denote the group generated by the elements enclosed, and whereθ, α, β are real numbers, α, β being in addition positive. Therefore, the group atany base point of the reduced FGM material groupoid coincides with the respectivesymmetry group, that is

N (X) = G(X) ∀x ∈ B.This means that, even if the material groupoid G(B) (the set consisting of materialisomorphisms and symmetries) is not transitive (i.e. B is not uniform), the reducedFGM material groupoid N (B) is, and it coincides with G(B) on the symmetrygroups. Thus, there is some kind of uniformity that generalizes the classical one.Finally, note that any G-structure related to N (B) will have a transversely isotropicstructural group as mentioned before.

Finally, note that we recover an analogue result to Theorem 5.8, which is alsotrue for fully isotropic FGM solids. If B is homosymmetrically relaxable, then fora unisymmetric and relaxable conguration K, the constitutive equation will beinvariant under the action of the structure group of the reduced N -stucture, relatedto the conguration K. In this case, the structure group will coincide through Kwith the symmetry group GK(X) at any point X in the domain of K. However, theconstitutive equation will not be independent of the point.


Functionally Graded Elastic Fluids

In the same way we have generalized the denition of elastic solids in section 5.4.3, weare going to give a new denition of elastic uids. Classically, an elastic uid is a uniformelastic material which posses a unimodular material structure, that is a U(3)-structure(see [146] for instance), even though there are smaller uid structures as the ones of uidcrystals (cf. [118]).

Denition 5.21. We will say that a functionally graded elastic material B is a function-ally graded uid (or a functionally graded uid crystal) if there is a volume form ρ onB invariant under material symmetries such that every point is uid (or, respectivelly, ifevery point is a uid crystal). The volume form is called a material form.

As in the case of functionally graded elastic solids, the following two conditions oncross-sections (U, σ) of the frame bundle FB,

σ(X)−1 · Gx · σ(X) ⊆ U(3) ∀X ∈ U ∀(U, σ) (5.23)

σ(X)−1 · τ(X) ∈ U(3) ∀X ∈ U ∩ V ∀(U, σ), (V, τ) (5.24)

characterize the uid material structure.Given a functionally graded elastic uid B, consider the unimodular groupoid U(B)

related to the volume form ρ (Example B.7). When two uid points have conjugatesymmetry groups, only the unimodular part of the conjugate transformation plays a rolein the conjugation. That is, if P is the transformation that conjugates these two groups,then the unimodular transformation P/detρ(P ) still realizes the conjugation.

Proposition 5.22. If B is a unisymmetric elastic uid, then

N 1(B) = N (B) ∩ U(B), (5.25)

where N 1(B) is the unimodular reduction of the FGM material groupoid.

Let B a uid crystal of rst kind (see [118, 154]), that is, an elastic uid as in 5.21such that, for each material point X ∈ B, the symmetry group G(X) may be representedfor some reference z at X by matrices of the form

A =

a b 0c d 0e f g

with det(A) = ±1. The normalizer in Gl(3) of this group of matrices is the set ofmatrices of the same form but with the restriction det(A) 6= 0. Therefore, when weintersect the normalizer with U(3) we obtain the original group of matrices. This meansthat N 1(X) = G(X) for every material point x ∈ B.

The latter example shows us how a uid material, which is not necessarily uniform,preserves uniformly the symmetry group structure across the body.

Appendix A

Multi-index properties

Given a function f : Rm −→ R, its partial derivatives are classically denoted

fi1i2···ik =∂kf

∂xi1∂xi2 · · · ∂xik.

When smooth functions are considered, their cross derivatives coincide. Thus, the orderin which the derivatives are taken is no longer relevant, but the number of times withrespect to each variable.

Another notation to denote partial derivatives is dened through symmetric multi-indexes (see [139]). A multi-index I will be an m-tuple of non-negative integers. The i-thcomponent of I is denoted I(i). Addition and subtraction of multi-indexes are denedcomponent-wise (whenever the result is still a multi-index), (I ±J)(i) = I(i)±J(i). Thelength of I is the sum |I| =

∑i I(i), and its factorial I! = ΠiI(i)!. In particular, 1i will

be the multi-index that is zero everywhere except at the i-th component which is equalto 1.

Keeping in mind the above denition, we shall denote the partial derivatives of afunction f : Rm −→ R by:

fI =∂|I|f

∂xI=

∂I(1)+I(2)+···+I(m)f

∂xI(1)1 ∂x

I(2)2 · · · ∂xI(m)

m

.

Thus, given a multi-index I, I(i) denotes the number of times the function is dierentiatedwith respect to the i-th component. The former notation should not be confused with thelatter one. For instance, the third order partial derivative ∂3f

∂x2∂x3∂x2(with f : R4 −→ R)

is denoted f232 and f(0,2,1,0), respectively.Here we present some simple, but useful, results on multi-indexes.

Lemma A.1. Given k integers 1 ≤ i1, . . . , ik ≤ m, with k ≥ 0 and m ≥ 1, dene thefunction

n(i1, . . . , ik) := Πkl=1Il(il) (n(∅) := 1), (A.1)

where Il := 1i1 + · · · + 1il ∈ Nm, for l = 1, . . . , k. We have that n is invariant underpermutations, that is, if π ∈ Σk is a permutation of k elements, then

n(i1, . . . , ik) = n(iπ(1), . . . , iπ(k)). (A.2)

Moreover, n(i1, . . . , ik) = Ik!.

137

138 APPENDIX A. MULTI-INDEX PROPERTIES

Proof. We proceed by induction on k. The cases k = 0 and k = 1 are trivial thus,let us suppose that the result is true for some integer k ≥ 1 to show that it is alsotrue for k + 1. Since n(i1, . . . , ik, ik+1) = Ik+1(ik+1) · n(i1, . . . , ik), by the hipotesys ofinduction, it sues to show that n(i1, . . . , ik−1, ik, ik+1) = n(i1, . . . , ik−1, ik+1, ik), which isequivalent to Ik+1(ik+1) · Ik(ik) = I ′k+1(ik) · I ′k(ik+1), where I ′k = 1i1 + · · · + 1ik−1

+ 1ik+1

and I ′k+1 = 1i1 + · · ·+ 1ik−1+ 1ik+1

+ 1ik .

Ik+1(ik+1) · Ik(ik) = (k−1∑l=1

δikil + δikik + δikik+1)(k−1∑l=1

δik+1

il+ δ

ik+1

ik+1)

=k−1∑l=1

δikil ·k−1∑l=1

δik+1

il+

k−1∑l=1

δikil +k−1∑l=1

δik+1

il+ δikik+1

·k−1∑l=1

δik+1

il

=k−1∑l=1

δik+1

il·k−1∑l=1

δikil +k−1∑l=1

δik+1

il+

k−1∑l=1

δikil + δik+1

ik·k−1∑l=1

δikil

= (k−1∑l=1

δik+1

il+ δ

ik+1

ik+1+ δ

ik+1

ik)(k−1∑l=1

δikil + δikik+1)

= I ′k+1(ik) · I ′k(ik+1)

Note that J ! = J(i) · I! for any pair (I, i) such that I + 1i = J .

Lemma A.2. 1. Let aI,iI,i be a family of real numbers indexed by a multi-indexI ∈ Nm and by an integer i such that 1 ≤ i ≤ m. Given an integer k ≥ 1, we havethat ∑

|I|=k−1

m∑i=1

aI,i =∑|J |=k

∑I+1i=J

aI,i. (A.3)

2. More generally, let aI1,I2I1,I2 be a family of real numbers indexed by two multi-indexes I1, I2 ∈ Nm. Given two integers k ≥ l ≥ 0, we have that∑

|I1|=l

∑|I2|=k−l

aI1,I2 =∑|J |=k

∑I1+I2=J|I1|=l

aI1,I2 . (A.4)

Proof. The proof is trivial when we realize that the sets (I, i) : |I| = k− 1, 1 ≤ i ≤ mand (I, i) : I + 1i = J, |J | = k are in bijective correspondence. For the general case,we shall consider the sets (I1, I2) : |I1| = l, |I2| = k − l and (I1, I2) : I1 + I2 =J, |I1| = l, |J | = k.

∑|I1|+|I2|=k

aI1,I2 =k∑l=0

∑|I1|=l

∑|I2|=k−l

aI1,I2 =∑|J |=k

∑I1+I2=J

aI1,I2 . (A.5)

139

Lemma A.3. Let aI,iI,i be a family of real numbers indexed by a multi-index I ∈ Nm

and by an integer i such that 1 ≤ i ≤ m. If J = 1i1 + · · ·+ 1ik ∈ Nm is a multi-index oflength k ≥ 0, we then have that

∑I+1i=J

aI,i =k∑l=1

1

J(il)· aJl,il , (A.6)

where Jl := 1i1 + · · ·+ 1il−1+ 1il+1

+ · · ·+ 1ik .

Proof. We proceed by induction on the dimension of the multi-indexes, m. The casem = 1 is clear thus, let us suppose that the result is true for m − 1 ≥ 1 to show thatit is also true for m. We rst note without lose of generality that we may suppose thati1 ≤ i2 ≤ · · · ≤ ik and that J(m) 6=. Otherwise, we could easily reorder the indexes andthe coordinates correspondingly before the computations and undo the changes at theend. ∑

I+1i=J

aI,i =∑

I+1i=Ji 6=m

aI,i + aJ−1m,m

=∑

I+1i=J

aI+J(m)1m,i + aJ−1m,m

where J = J − J(m)1m

=

k−J(m)∑l=1

1

J(il)· aJl+J(m)1m,il

+ aJ−1m,m

where we have applied the hypothesis of induction

=

k−J(m)∑l=1

1

J(il)· aJl,il +

k∑l=k−J(m)+1

1

J(il)· aJl,il

Lemma A.4. Let J ∈ Nm be a non-zero multi-index. We have that∑I+1i=J

I(i) + 1

|I|+ 1= 1. (A.7)

Proof.

1 =m∑i=1

J(i)

|J |=∑

I+1i=J

J(i)

|J |=∑

I+1i=J

I(i) + 1

|I|+ 1

Lemma A.5. Let aJJ be a family of real numbers indexed by a multi-index J ∈ Nm.Given a positive integer l ≥ 1, we have that

∑|J |=l

aJ =∑|I|=l−1

m∑i=1

I(i) + 1

|I|+ 1aI+1i , (A.8)

140 APPENDIX A. MULTI-INDEX PROPERTIES

Proof. ∑|J |=l

aJ =∑|J |=l

( ∑I+1i=J

I(i) + 1

|I|+ 1

)aJ

=∑|J |=l

∑I+1i=J

I(i) + 1

|I|+ 1aI+1i

=∑|I|=l−1

m∑i=1

I(i) + 1

|I|+ 1aI+1i .

Lemma A.6. Let aJJ be a family of real numbers indexed by a multi-index J ∈ Nm.Given a positive integer k ≥ 1, we have that∑

|J |=k

aJ =∑

1≤j1,...,jk≤m

Jk!

k!aJk , (A.9)

where Jk := 1j1 + · · ·+ 1jk .

Proof. We proceed by induction on k. The case k = 1 is trivial thus, let us suppose thatthe result is true for some integer k ≥ 1 to show that it is also true for k + 1.∑

|J |=k+1

aJ =m∑

jk+1=1

∑|J |=k

J(jk+1) + 1

k + 1aJ+1jk+1

=m∑

jk+1=1

∑1≤j1,...,jk≤m

Jk!

k!

Jk(jk+1) + 1

k + 1aJk+1jk+1

=∑

1≤j1,...,jk+1≤m

Jk+1(jk+1) · Jk!(k + 1)!

aJk+1

Lemma A.7. LetaJ , b

JJbe a family of real numbers indexed by a multi-index J ∈ Nm.

Given an integer l ≥ 1, we have that∑|J |=l

bJaJ =∑|I|=l−1

m∑i=1

I(i) + 1

|I|+ 1(bI+1i +QI,i)aI+1i , (A.10)

whereQI,i

I,i

is a family of real numbers such that for any multi-index J ∈ Nm (with|J | ≥ 1) we have that ∑

I+1i=J

I(i) + 1

|I|+ 1QI,i = 0. (A.11)

Proof. ∑|J |=l

bJaJ =∑|J |=l

( ∑I+1i=J

I(i) + 1

|I|+ 1

)bJaJ

=∑|J |=l

∑I+1i=J

I(i) + 1

|I|+ 1(bI+1i +QI,i)aI+1i .

Appendix B

Groupoids and G-structures

B.1 Groupoids

Groupoids are a generalization of groups; indeed, they have a composition law withrespect to which there are some identity elements and every element has an inverse. Fora good reference on groupoids, the reader is refered to Mackenzie [117].

Denition B.1. Given two sets Ω and M , a groupoid Ω over M , the base, consists ofthese two sets together with two mappings α, β : Ω→M , called the source and the targetprojections, and a composition law satisfying the following conditions:

1. The composition law is dened only for those η, ξ ∈ Ω such that α(η) = β(ξ) and,in this case, α(ηξ) = α(ξ) and β(ηξ) = β(η). We will denote Ω∆ ⊂ Ω × Ω the setof such pairs of elements.

2. The composition law is associative, that is ζ(ηξ) = (ζη)ξ for those ζ, η, ξ ∈ Ω suchthat each member of the previous equality is well dened.

3. For each x ∈M there exists an element 1x ∈ Ω, called the unity over x, such that

(a) α(1x) = β(1x) = x;

(b) η · 1x = η, whenever α(η) = x;

(c) 1x · ξ = ξ, whenever β(ξ) = x.

4. For each ξ ∈ Ω there exists an element ξ−1 ∈ Ω, called the inverse of ξ, such that

(a) α(ξ−1) = β(ξ) and β(ξ−1) = α(ξ);

(b) ξ−1ξ = 1α(ξ) and ξξ−1 = 1β(ξ).

The groupoid Ω will be said transitive if, for every pair x, y ∈M , the set of elements thathave x as source and y as target, i.e. Ωx,y = α−1(x) ∩ β−1(y), is not empty.

A subset Ω′ ⊂ Ω is said to be a subgroupoid of Ω over M if itself is a groupoid overM with the composition law of Ω.

The elements ofM are often called objects and those of Ω arrows due to their graphicalinterpretation as we may see in the Figure B.1 or in the example B.2. By the verydenition of groupoids, the unity over an object and the inverse of an arrow are unique.Note also that Ωx,x is a group and the unity 1x is the group identity.

141

142 APPENDIX B. GROUPOIDS AND G-STRUCTURES

Figure B.1: The arrow picture.

Example B.2 (The trivial groupoid). Let M denote any non-empty set. The Cartesianproduct M ×M is trivially a groupoid over M . The source of an arrow (x, y) is x andthe target y, and the composition (y′, z) · (x, y) is (x, z) if and only if y′ = y.

Example B.3 (The action groupoid). Now, let G be a group acting on the left on M .Then the product G×M is a groupoid over M with the following structural maps:

• the source, α(g, x) = x;

• the target, β(g, x) = g · x;

• and the composition law, (h, y) · (g, x) = (h · g, x) if and only if y = g · x.

With these considerations, the unity over an element x ∈M and the inverse of an arrow(g, x) ∈ G×M are respectively given by 1x = (e, x) and (g−1, g · x), where e ∈ G denotesthe identity and g−1 the inverse of g.

Proposition B.4. Let Ω be a groupoid over a setM . Then, given three points x, y, z ∈Msuch that they can be connected by arrows, we have the relation

Ωx,z = g · Ωx,y = Ωy,z · f, ∀g ∈ Ωy,z, ∀f ∈ Ωx,y; (B.1)

in particular,Ωy,y = g · Ωx,x · g−1, ∀g ∈ Ωx,y. (B.2)

For the moment, we have only algebraic structures on groupoids. Let us endow themwith dierential structures.

Denition B.5. We say that a groupoid Ω over M is a dierential groupoid if thegroupoid Ω and the base M are equipped with respective dierential structures suchthat:

1. the source and the target projections α, β : Ω→ M are smooth surjective submer-sions;

2. the unity or inclusion map i : x ∈M 7→ 1x ∈ Ω is smooth;

3. and the composition law, dened on Ω∆, is smooth.

B.1. GROUPOIDS 143

Additionally if Ω is transitive, then we call it a Lie groupoid.A subgroupoid Ω′ of a dierential (or Lie) groupoid Ω which is in turn a dierential

groupoid with the restricted dierential structure is called a dierential subgroupoid (resp.Lie subgroupoid).

Note that the condition (1) in Denition B.5 implies that the αβ-diagonal Ω∆ is anembeded submanifold of Ω× Ω, and then (3) makes sense. Ver Eecke showed (cf. [117])that, even with more relaxed conditions, the inverse map ξ ∈ Ω 7→ ξ−1 ∈ Ω is smooth, andtherefore a dieomorphism. In fact, there is a more general way to dene groupoids andsubgroupoids (dierentiable or not) as the reader may nd in [117], but for our purposesthese denitions will be sucient.

Example B.6 (The frame groupoid). Let M be a smooth manifold with dimension n andconsider the space of linear isomorphisms between tangent spaces to M at any pair ofpoints, namely

Π(M) =⋃

x,y∈M

Iso(TxM,TyM). (B.3)

This set is called the frame groupoid of M and, in fact, it is a Lie groupoid over M , aswe are going to show.

First of all, we must give a manifold structure to Π(M). Let (U, φ) and (V, ψ) be twocharts of M and consider the map given by

χ : W −→ φ(U)×Gl(n)× ψ(V )

A 7−→ (xi, Aji , yj)

(B.4)

where Gl(n) denotes the general linear group on Rn,

W =⋃

x∈U,y∈V

Iso(TxM,TyM) and A

(∂

∂xi

)= Aji

∂

∂yj. (B.5)

By means of the induced chart (W,χ) we endow Π(M) with a dierential structure ofdimension 2n+ n2.

The structural maps are given in the following way:

• the source and the target projections: if A ∈ Iso(TxM,TyM), then α(A) = x andβ(A) = y;

• the composition law is the natural composition between isomorphisms when it isdened;

• and the inclusion: if x ∈ M , then the unity 1x over x is the identity map ofGl(TxM) = Iso(TxM,TxM).

These maps dene clearly a groupoid over M and, through (B.4) and (B.5), they aresmooth for the dierential structure naturally induced from the one of M .

Example B.7 (The unimodular groupoid). Let M be an orientable smooth manifold ofdimension n and let ρ be a volume form on it (in a more general case, without theassumption of orientation, we can consider a volume density). We can use ρ to dene adeterminant function over the frame groupoid Π(M) by the formula:

ρ(A · v1, . . . , A · vn) = detρ(A) · ρ(v1, . . . , vn) ∀A ∈ Π(M), (B.6)


where v1, . . . , vn ∈ Tα(A)M . Now, it is easy to check that the set of unimodular transfor-mations

U(M) = det−1ρ (−1,+1), (B.7)

which is called the unimodular groupoid, is a transitive subgroupoid of Π(M). In fact,it is a Lie subgroupoid of Π(M), since detρ is a smooth submersion and thus U(M) is aclosed submanifold.

Example B.8 (The orthogonal groupoid). Let (M, g) be a Riemannian manifold of dimen-sion n and consider the space of orthogonal linear isomorphisms between tangent spacesto M at any pair of points, namely

O(M) =⋃

x,y∈M

O(TxM,TyM). (B.8)

This set is called the orthogonal groupoid of M and, with the restriction to it of thestructure maps of the frame groupoid Π(M), O(M) is a subgroupoid of Π(M). SinceO(M) is dened by closed and smooth conditions, namely

O(M) =A ∈ Π(M) : A−1 = AT

,

this set is a closed submanifold of Π(M), and thus a Lie subgroupoid.Furthermore, the orthogonal groupoid O(M) is also a Lie subgroupoid of the unimod-

ular groupoid U(M) related to the Riemannian density induced by the metric.

Denition B.9. Let Ω be a groupoid over M ; then the normalizoid of a subgroupoid Ωof Ω is the set dened by

N(Ω) =g ∈ Ωx,y : Ωy,y = g · Ωx,x · g−1, x, y ∈ B

. (B.9)

From the denition, it is obvious that a subgroupoid Ω of a groupoid Ω is also asubgroupoid of its normalizoid N(Ω) which is, in turn, a subgroupoid of the ambientgroupoid Ω.

Note that the group over a base point in the normalizoid is the normalizer of thegroup over this point in the subgroupoid, that is

(N(Ω))x,x = N(Ωx,x), (B.10)

which explains the used terminology. The dierence between a subgroupoid and itsnormalizoid can be huge. For instance, given a transitive groupoid Ω over a set M ,consider its base groupoid, that is the subgroupoid consisting of the groupoid unities:

1(Ω) = 1x : x ∈M . (B.11)

Then, the normalizoid of 1(Ω) in Ω is the whole groupoid Ω. From now on, we will focuson subgroupoids of the frame groupoid over a manifold and we will see how to reduce thenormalizoid of a subgroupoid whenever an extra structure is avaible on the base manifold.

First of all, recall that there exists a unique decomposition of a linear isomorphisminto an orthogonal part and a symmetric one. More precisely, let F : E −→ E ′ bea linear isomorphism between two inner product vector spaces E and E ′. There exist

B.1. GROUPOIDS 145

an orthogonal map R : E −→ E ′ and positive denite symmetric maps U : E −→ E,V : E ′ −→ E ′ such that:

F = R · U and F = V ·R. (B.12)

As we have mentioned, each of these decompositions is unique and they are called the leftand right polar decompositions of F , respectively; the orthogonal part R will be denotedby F⊥.

Proposition B.10. Let Ω be a (transitive) subgroupoid of the frame groupoid Π(M) ofa Riemannian manifold (M, g). Denote by Ω the set of the orthogonal part of elementsof Ω, that is

Ω =F⊥ : F ∈ Ω

. (B.13)

Then Ω is a (transitive) subgroupoid of the orthogonal groupoid O(M). We call Ω theorthogonal reduction of Ω (or the reduced groupoid, for the sake of simplicity).

Proof. In order to show that Ω is a subgroupoid of O(M), we only have to check that itis a groupoid over M with the restriction of the structure maps of Π(M), which is clearonce we note that for any three linear isomorphisms F1, F2, F3, such that F3 = F2 ·F1, wehave by the uniqueness of the polar decomposition that F⊥3 = F⊥2 · F⊥1 .

Note that the orthogonal reduction of a normalizoid is not necessarily a subgroupoidof the original one.

Proposition B.11. In the hypotesis of Proposition B.10, if Ω is such that, for everybase point x ∈ M , Ωx,x is a subgroup of Ox,x(M) (the orthogonal group at x), thenthe orthogonal reduction of the normalizoid of Ω coincides with the intersection of theorthogonal groupoid and the normalizoid itself, i.e.

N (Ω) = N (Ω) ∩ O(M). (B.14)

Proof. The inclusion N (Ω) ⊃ N (Ω) ∩ O(M) is clear and, from the above PropositionB.10, we have N (Ω) ⊂ O(M), thus we only need to show that N (Ω) ⊂ N (Ω). LetR ∈ Nx,y(Ω), then there exist a linear isomorphism F ∈ Nx,y(Ω) such that F⊥ = R.Since F conjugates the orthogonal subgroups Ωx,x and Ωy,y, so does its orthogonal part(cf. [81], Lemma A.2). Hence, R ∈ Nx,y(Ω) and N (Ω) ⊂ N (Ω) ∩ O(M).

Similar results can be given whenever M is equipped with a volume form.

Proposition B.12. Given a smooth manifold M , suppose it is endowed with a volumeform (or density) ρ. If Ω denotes a (transitive) subgroupoid of the frame groupoid Π(M),then the set

Ω1 = Ω/detρ, (B.15)

is a (transitive) subgroupoid of the unimodular groupoid U(M) associated with ρ and itwill be called the unimodular reduction of Ω.

Even more, if Ω is such that, for every base point x ∈ M , Ωx,x is a subgroup ofUx,x(M) (the unimodular group at x), then the unimodular reduction of the normalizoidof Ω coincides with the intersection of the unimodular groupoid and the normalizoid itself,i.e.

N 1(Ω) = N (Ω) ∩ U(M). (B.16)


B.2 G-structures

Lie subgroupoids of the frame groupoid of a manifold are closely related to anothergeometric object: G-structures, which are a particular case of ber bundles. For a com-prehensive reference related to principal ber bundles and G-structures see [84, 110, 111].We give here their denition and some results about the interconnection with groupoids.

Denition B.13. Given two manifolds P,M and a Lie group G, we say that P is aprincipal bundle over M with structure group G if G acts on the right on P and thefollowing conditions are satised:

1. the action of G is free, i.e. the fact that ua = u for some u ∈ P implies a = e, theidentity element of G;

2. M = P/G, which implies that the canonical projection π : P −→ M is dieren-tiable;

3. P is locally trivial, i.e. P is locally isomorphic to the product M ×G, which meansthat for each point x ∈ M there exists an open neighborhood U and a dieomor-phism Φ : π−1(U) −→ U×G such that Φ = π×φ, where the map φ : π−1(U) −→ Ghas the property φ(ua) = φ(u)a for all u ∈ π−1(U), a ∈ G.

A principal bundle is commonly denoted by P (M,G), π : P −→M or simply by P , whenthere is no ambiguity. The manifold P is called the total space, M the base space, G thestructure group and π the projection. The closed submanifold π−1(x), with x ∈ M , iscalled the ber over x and is denoted Px; if u ∈ P , Pπ(u) is called the ber through u andis denoted Pu. The maps given in (3) are called (local) trivializations.

It should be remarked that a similar denition can be given for left principal bundlesusing left actions.

Notice that any ber Px is dieomorphic to the structure group G, but not canonicallyso. On the other hand, if we x u ∈ Px, then Pu = uG. We may visualize a principalber bundle P (M,G) as a copy of the structure Lie group G at each point of the basemanifold M in a dientiable way as it is stated by the trivialization property (3).

An elementary example of principal bundle is the frame bundle FM of a manifoldM .This manifold consists of all the reference frames at all the point ofM . The frame bundleFM is a principal bundle over M with structure group Gl(n), where n is the dimensionof M . As it is obvious, the canonical projection π sends any frame x ∈ FM to the basepoint x ∈ M where it lies. The right action of Gl(n) over M is dened in the followingway:

R : FM ×Gl(n) −→ FM(z, a) 7−→ Raz = z · a = (ajivj),

(B.17)

where (aji ) is the matrix representation of a ∈ Gl(n) in the canonical basis of Rn and (vi)is the ordered basis given by z ∈ FM .

Denition B.14. Let P (M,G) and Q(M,H) be two principal bundles such that Q isan embedded submanifold of P and H is a Lie subgroup of G. We say that Q(M,H) isa reduction of the structure group G of P if the principal bundle structure of Q(M,H)comes from the restriction of the action of G on P to H and Q. In this case, we call Qthe reduced bundle.

B.2. G-STRUCTURES 147

Consider the following (non rigorous) construction: take a principal bundle P (M,G),shrink its structure group to a Lie subgroup H of G, x an element u ∈ P in each breof the bundle and apply the action of H to each of these chosen elements; this gives us asubset Q ⊂ P . The obtained set Q is a reduced bundle when the selection of the u's ismade smoothly and with certain compatibility.

Denition B.15. Let M be an n-dimensional smooth manifold and G a Lie subgroupof Gl(n); then a G-structure G(M) is a G-reduction of the frame bundle FM .

Note that there may exist dierent G-structures with the same structure group. Asan example of G-structure, consider a Riemannian manifold (M, g). The set of orthonor-mal references of FM gives us an O(n)-structure. In fact, any O(n)-structure on M isequivalent to a Riemannian structure (see [84]).

Now let us introduce two results from [118] that show how a G-structure may arisefrom a Lie groupoid.

Proposition B.16. Let Ω be a Lie groupoid over a smooth manifold M with source andtarget projections α and β, respectively. Given any point x ∈M , we have that:

1. Ωx,x = α−1(x) ∩ β−1(x) is a Lie group and

2. Ωx = α−1(x) is a principal Ωx,x-bundle over M whose canonical projection is therestriction of β.

Given a smooth manifold M of dimension n, any reference z ∈ FM (at a pointx ∈ M) may be seen as the linear mapping ei ∈ Rn 7→ vi ∈ TxM , where (e1, . . . , en) isthe canonical basis of Rn and (v1, . . . , vn) the basis of TxM dened by z.

Theorem B.17. Suppose that M is a smooth n-dimensional manifold and Ω is a Liesubgroupoid of the frame groupoid Π(M). If α and β denote the respective source andtarget projections of Ω, then we have that for any point x ∈ M and any frame referencez ∈ FM at x:

1. Gz = z−1 · Ωx,x · z is a Lie subgroup of Gl(n) and

2. the set Ωz of all the linear frames obtained by translating z by Ωx, that is

Ωz = gx,y · z : gx,y ∈ Ωx , (B.18)

is a Gz-structure on M .

Once the reference z is xed, the linear frames that lie in the Gz-structure are calledadapted or distinguished references.

Even though the frame groupoid (and hence each of its subgroupoids) acts on the lefton the frame bundle of the base manifold, the structural group that arises from a framesubgroupoid acts naturally on the right on any of the induced G-structures:

zy · gzx = (gx,y · zx) · (z−1x · gx,x · zx) = gx,y · gx,x · zx = g′x,y · zx = z′y, (B.19)

where zx ∈ FxM , zy ∈ (Ωzx)y, gzx ∈ Gzx , gx,y ∈ Ωx,y and so on.


Remark B.18. It is readily seen from equation (B.2) that two G-structures that comefrom the same Lie groupoid are equal if and only if they have a reference in common,

Ωz1 = Ωz2 ⇔ Ωz1 ∩ Ωz2 6= ∅. (B.20)

Here equal means that the two G-structures are the same as sets and they have thesame structure groups. By the above statement, given two G-structures Ωz1 and Ωz2

induced by a Lie groupoid Ω, we can suppose without loss of generality that z1 and z2

are linear frames at the same base point. Thus, it is easy to see that their respectivestructure groups Gz1 and Gz2 are conjugate; more precisely:

Gz2 = z−12 z1 ·Gz1 · z−1

1 z2. (B.21)

In short, given a Lie subgroupoid Ω of Π(M), the frame bundle FM is the disjoint unionof G-structures related to Ω by Theorem B.17. Moreover, they have conjugate groupstructures and one of these G-structures may be transformed to another by means ofany element g ∈ Gl(n) that conjugates their structural groups. Hence, modulo thesetransformations, a G-structure related to a Lie subgroupoid Ω of Π(M) is unique, whichis clear since Ω is xed.

A natural question is whether Theorem B.17 has a converse. Given a G-structure, itseems reasonable to be able to choose dierentially isomorphisms that transform adaptedreferences to their counterparts.

Theorem B.19. Let ω be a G-structure over an n-dimensional smooth manifold M .Then the set of linear isomorphism that transforms distinguished frames into distinguishedframes, that is the set

Ω =A ∈ Π(M) : Az ∈ ω, z ∈ ωα(A)

, (B.22)

where Π(M) is the frame groupoid ofM and α the source projection, is a Lie soubgroupoidof Π(M). Furthermore, for any reference frame z ∈ ω, the G-structure associated to Ωand given by Theorem B.17 coincides with ω, i.e.

Ωz = ω and Gz = G. (B.23)

Proof. The set dened by equation (B.22) is obviously a transitive subgroupoid of Π(M).It remains only to show that it is a dierential groupoid with the restriction of thestructural maps. Given two local cross-sections (U, σ) and (V, τ) of ω, consider the set ofisomorphisms in Ω with source in U and target in V , namely

ΩU,V = α−1(U) ∩ β−1(V ), (B.24)

where α and β are the restrictions to Ω of the source and the target projections of Π(M).Given an isomorphism A ∈ ΩU,V , let x = α(A) ∈ U and y = β(A) ∈ V . If we denotethe components of the ordered bases σ(x) and τ(y) by (σi(x)) and (τj(y)) respectively,we have that there exist coecients Aji such that

Aσi(x) = Ajiτj(y). (B.25)

Since σ(x) = (σi(x)) is a linear frame at x in ω, Aσ(x) = (Ajiτj(y)) is a linear frameat y in ω too. But τ(y) = (τj(y)) is also a linear frame at y in ω, thus a = (Aji ) must

B.2. G-STRUCTURES 149

necessarily be an element of the structure group G. This consideration being made, wedene the coordinate chart Φσ,τ by

Φσ,τ : ΩU,V −→ U ×G× VA 7−→ (x, a, y)

. (B.26)

Given a covering of M by local sections of ω, say Σ, the atlas

(ΩU,V ,Φσ,τ ) : (U, σ), (V, τ) ∈ Σ (B.27)

denes a smooth structure on Ω, from which it is a straightforward computation to showthat the projections α and β and the composition law are smooth.

Remark B.20. The result we have just proved, toghether with Theorem B.17, shows theequivalence between Lie subgroupoids of Π(M) and reductions of the frame bundle FM .In fact it is still true for principal bundles in general: by Proposition B.16 we are able toassociate some principal bundles to a groupoid and, given a principal bundle P (M,G),the set of maps φx,y : Px −→ Py such that φx,y(u · g) = φx,y(u) · φ(g), for a suitable groupisomorphism φ : G −→ G, is a Lie groupoid related to P by Proposition B.16.

Denition B.21. A G-structure G(M) over a manifoldM is said to be integrable if thereexists an atlas (Uα, φα)α∈A of the base manifold, such that the induced cross-sectionsσα(x) = (Txφα)−1 take values in G(M).

By the very denition, if a G-structure is integrable, the same happens to all itsconjugate G-structures.

Theorem B.22. A G-structure over a manifold M with dimension n is integrable if andonly if it is locally isomorphic to the standard G-structure of Rn, that is, to Rn ×G.

The following result will be useful in the next section.

Lemma B.23. Let M be a manifold. If Ω and Ω are two subgroupoids of the framegroupoid Π(M), then their intersection Ω := Ω ∩ Ω is again a subgroupoid of Π(M) (andof Ω and Ω). Furthermore, if they are Lie groupoids, then we have the following relations:

Ωz = Ωz ∩ Ωz and Gz = Gz ∩ Gz, (B.28)

where z ∈ FM is a xed frame and Ωz, Ωz, Ωz, Gz, Gz and Gz are the respective G-structures and structural groups.

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